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.
You might be able to use the NiCd charger to put charging current into the battery. It's the part where you "wait until finished" that the problem arises. The charge termination schemes (when to automatically stop charging) will be different. For lead-acid, probably switching from current-limit mode to voltage-limit mode at some maximum safe charging voltage. For NiCd/NiMH, watching the voltage for where it goes "over the hill" from increasing to decreasing, and/or the temperature rises quickly. Plus some other safety features, such as timeout, over/under-voltage, over/under-temperature, etc.
If you put the wrong chemistry on the charger it may never see what it's looking for to cease charging, because the battery behaves differently (voltage graphed over time has a different pattern). It could keep charging literally until the battery explodes. If you understand what the battery requires, and what the charger does, you might be able to use it anyway, to the extent you can manually observe and terminate charging as appropriate. You're risking destroying the battery, but in some cases getting a partial charge right now is more valuable than a longer lifespan later.
Your stated goal is measuring the capacity of the battery, rather than, say, getting enough charge back into your car battery so you can start the car and drive it around to finish recharging. If all you wanted to do was get "some" charge into the battery, considering you already know it's discharged, you could probably let it charge from your 2A source for "a while" and disconnect it manually after some time considerably less than (64A-hr/2A)=32hr, say 8-12 hours, watching with a voltmeter that the voltage doesn't exceed the maximum safe voltage and disconnecting early if it does. For your purposes of getting the battery fully charged and discharging to measure capacity, you'd want a proper charge termination scheme, which you don't have.
Best Answer
Frankly, if you want to use rechargeable batteries, I'd suggest to go for Li-ion batteries. A single-cell (3.7V nominal) or a double-cell (7.4V nominal) sound like good choices. For the single-cell version, you'd need a DC-DC boost step-up converter to get 5V, for the double-cell you can either use a "normal" dissipative power regulator (something similar to the 7805, but with a lower dropout voltage, since the minimum output voltage of a double-cell Li-ion battery is about 6V, and that's not enough for the 7805), or, if power loss is an important factor (I'd guess it is), you should use a DC-DC buck step-down converter. Also, if you don't want to prematurely kill your Li-ion batteries, you should use a power converter which shuts down when the minimum input voltage is reached (3V for a single-cell, 6V for a double-cell).
So I'd look for a battery with the proper capacity (you should do a rough calculation for the expected minimum milliamp-hours: Pout / efficiency / battery voltage * run time), and a circuit designed for converting from that battery into 5V. Ebay is for example full of these components (both Li-ion batteries and circuits designed for this purpose).