Its possible but hard to answer without knowing the charger topology.
NiCd can be handle very high charge/discharge currents. Such charge currents could damage a NiMH battery.
Now it MAY be OK, not all NiCd chargers are of the high current/peak detect variety, sometimes called "Rapid Charge".
Some are simply timer based in which case they will very likely undercharge a NiMH battery.
Still other NiCd chargers are "trickle chargers" made to slow charge overnight, such a charger would work with NiMH but would likely take a very long time to charge since NiMH batteries generally have a much higher capacity.
Yet another charger type is the "smart charger" which monitors the battery as it charges and adjusts its charge rate, stopping or going into a trickle mode when the battery is determined to be full. The "full" trigger and charge profiles are different between NiCd and NiMH so I would assume this type would need to be modified (possibly with new firmware) to handle NiMH batteries.
EDIT: Ancillary question: are you sure switching battery chemistry is a good idea for this application? The mAh rating doesn't tell the entire story of battery life. If the device is high current then a NiCd battery may last much longer than a NiMH battery even though the NiMH battery likely has much higher mAh rating.
In domestic use LiIon (Lithium Ion) batteries are, all things considered, MORE dangerous than "lead acid" batteries, not less dangerous. But both are "reasonably safe" [tm] when used properly.
The advice that you linked to above is actually titled "What precautions are needed when charging a car battery in an apartment?" and that is quite different than charging a car battery at home.
Specifically, a car battery is a one of a range of variants of lead acid batteries and contains liquid acid and while it has plugged vents and fillers it is not "sealed" in any adequate manner. Under certain conditions which are reasonably liable to be encountered in normal charging it may liberate either acid fumes or Hydrogen gas, or both. If it is charged in a car or outside it is unlikely to cause many problems.
Lithium Ion batteries when being charged do not usually liberate hydrogen or release electrolyte. Both are possible, but only if a damaged or incorrect charger is used. In exchange for not doing these things. instead they occasionally catch fire and "explode" - actually not a true explosion.
Each 18650 cell in a typical laptop battery contains the energy of about 12 high energy load '44 magnum' shells or about 24 "standard" .44 Magnum rounds, and that's just the electrical energy. (The energy from combustion can exceed 300 kJ, or over a hundred .44 Magnum rounds!). An entry level netbook has 3 such cells and a top notebook PC may have 9 or even 12 of these. This can be released in about 10 seconds with flame and 'lots of heat'. The standard industry term for this, somewhat tongue in cheek is "Vent with flame". The fact that a standard industry term exists indicates that it's a well known problem. The "melt down" mode can be triggered by charging to too high a voltage, discharging excessively and then charging normally, charging at an excessive current rate, puncturing them, or in selected cases* giving them a slightly harder than usual knock which causes internal parts to short together and - Wow!. (* This is obviously a manufacturing fault but has happened in a number of independent cases due to too tight manufacturing tolerances and substandard assembly).
All that said - given how many there are, LiIon batteries have proved to be very safe considering how many there are and how they are treated. Despite the horror stories I have never seen a real world "vent with flames" event.
Added 7 years on: I've now seen one. An old netbook with a dead LiIon battery pack was left connected to a power supply for several days. One evening it sudeenly burst into magnificent melt-down. Flame, smoke, ... - urgent defenestration of the battery save the netbook - and the couch was "not badly damaged". Had we not been in the room we'd have had a house fire.
Lead acid batteries do not have an equivalent failure mode to "vent with flame"
However, drop a spanner, vacuum cleaner metal suction tube or similar across the terminals of a car battery and you'll get immense energy release. The battery may be damaged by such treatment but usually won't explode. Charge them too fast or too long and Hydrogen gas will be produced and WILL leak out and can form a flammable or explosive mix in confined spaces. Battery acid from other than fully sealed lead acid batteries seems to be special Houdini grade - skilled at escaping in many unexpected instances. If you carry the battery and your arms itch, Wash them NOW. Next time you wash your clothes small holes with brown edges may appear(Ask me how I know). Charge them inside at above gassing point and you may be sorry. I was :-).
BUT Lead acid are also very safe as long as they are used as intended. Charging a wet plate lead acid car battery "inside" at home is not included in "as intended".
Some excellent related links, recommended by Nick Alexeev:
MPower UK - Lithium Battery Failures
SE EE - Why is there so much fear surrounding LiPo batteries? - good question and eleven good to great answers.
Best Answer
cIt's complex, and some of the answers are "soft" and some of you assumptions are (reasonably enough) inexact.
LiIon cells are frequently measured in mAh capacity.
LiIon batteries (1 or more cells) often have mWh and mAh markings.
Neither is a certain measure of what a user will receive.
Both are useful.
(cf George Box's "All models are wrong, some models are useful")
A LiIon cell have a mean voltage of 3.6 to 3.7V.
If you multiply the mAh rating x 3.6 or 3.7 you get the Wh capacity.
The true Ah capacity of ANY battery depends on usage profile - constant current, constant power, pulsed drain of X amps for xx ms every xxx ms etc.
LiIon is relatively close in mAh ratings across its capacity range compared to some other chemistries.
A LiIon cell in low to medium power applications (eg total discharge times of hours) has Vout of 4.2 V o/c fully charged and say 3V fully discharged at light or typical currents.
Vmean is say 3.6V.
If you discharge at constant power of say 10W then
at 4.2V I = P/V ~= 2.4A and
at 3V I = P/V = 10/ 3 = 3.33 A.
ie a change of 2.5:3.3 ~= 0.75:1 or 1.33.
If a mAh rating is used then a say 6Ah cell will give
t = Ah/I = 6/3.3 or 6/2.5
= 1.8 hours or 2.4 hours at the current extremes, and in practice somewhere in between.
The same cell will probably be rated as Wh = Ah x 3.6V = 6 x 3.6 = 21.6 Wh.
Run time = Wh/P = 21.6/10 = 2.6 hours.
In this case the true run time will probably not exactly match and of the above but is liable to be near or maybe above the 2.6 hour figure.
If discharge was at constant current different calculations apply.
Say we set 10W at 3.6V =~ 2.75A
At 4.2 V, P = V x I = 4.2 x 2.75 = 11.6 W
At 3V, P = 3 x 2.75 = 8.25W
Ah rating gives discharge life of t = Ah/I = 6/2.75 = 2.18 hours
Wh rating gives time of t = Wh/P_V3.6 = 21.6 / 10 = 2.16 hours.
Close enough.
Neither may be correct.
As you draw more current from a LiIon (or any other) cell the voltage will drop. Terminal voltage will depend on state of charge, current, capacity, past history, ... . A voltage is lower or much lower at very high drain, the chosen termination voltage will affect the apparent mAh capacity.
And more
So - a LiIon battery Wh rating from a reputable supplier is the approximate Wh achieved under typical use in a typical application. A battery in a laptop may (or may not) be rated differently than one in an eg power wall, or electric vehicle or power tool.
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More liable to be true.
NimH have a far flatter discharge voltage curve across their capacity range.
Usually we take Vmin = 1V or even higher.
Vavg may be around 1.2V at low to medium loads and l.2 - 1.15V as load increases - and lower under very heavy loads.
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Largely true.
Datasheets with extensive to extremely extensive information are available from reputable manufacturers. Those who know the fine detail of their requirements and wish to design for an application are generally well served.
Those who wish to pop in a new set of batteries are passed on to the marketing department.
Top AA Alkalines are nowadays about on par mAh wise with the top NimH AA cells.
Actual capacity can depend very substantially on the application and conditions. No/low/medium/high currents, pulsed versus intermittent versus continuous loads and more alter the results.
If you are an end user and you want a well performing battery then "long lasting" and "longest lasting" are less important than the brand name. mAh or Wh ratings would be a guide, but no better than the brand name for most users.
For longest lasting top energy AA cells Eveready Lithium primary cells seem currently to be "it". Other reputable brands products will not be far behind They have a very flat discharge curve at usefully higher voltages than Alkalines across the discharge range.
I personally never buy top brand Alkalines as I have found that "trustable"* volume-sold rebranded Alkalines provide better energy capacity per $.
Usually those sold by larger chain stores or major outlets.
MUST weigh >= 20 grams. 23g better. More again better still.
"New" Alkaline Vo/c:
1.65V+ New new.
1.6 - 1.65 V - 0-6 months. Maybe 1 year since manufacture.
1.55 - 1.6V - 12 -24 months old. Maybe more.
Under 1.55V - NOT Alkaline or very old.