NiMH battery for Electric Bike

NiMH cells start at about 1.5 V right when fully charged, drop to about 1.2 V most of their discharge life, and are pretty much empty at 900 mV. Stopping there is usually safe. 800 mV is where you definitely want to stop to avoid damage. There is so little energy left at that point that there is no benefit in draining the cell further anyway.

You may think therefore that your 10 cell pack can therefore be discharged to 9 or 8 V, but unfortunately not. There will always be some imbalance between cells. If you can measure individual cells, you can go until the lowest cell hits 800 mV, but then stop right away. The cell with the least capacity will get there first. Once it does, its internal resistance goes up and further current causes the voltage to drop rapidly, causing permanent damage.

10 NiMH cells really should not be put in series without a way to at least measure individual cells. If you designed the battery pack, then you need to fix this. If someone else did, then they are not trustworthy and it would be a good idea to dump them and find someone that knows what they are doing. With 10 cells, it's hard to pick a reasonable stopping point because possible imbalance between cells could be significant, especially after a few charge/discharge cycles. Maybe use 1.1 V average per cell, but this is really not a good way to deal with a 10-cell pack.

You will have the same problem with charging. You will have to use relatively low charge current, like maybe C/4 until you think the first cell is near full, then maybe a C/10 or so to trickle charge for a couple of hours so the other cells hopefully catch up. Again, the right answer is to not get yourself into this mess in the first place. Packs with this many cells need to have individual cells measured at least, and the best way is to have some charge balancing circuitry. This shunts some charge current around the full cells so that they don't get overcharged while the less full cells catch up. Of course this requires measuring individual cells to know when to enable the shunts per cell.

Again, work with someone or a company that actually knows what they are doing. This sort of thing is a lot more complex than it appears at first glance.

LiPo is MUCH easier to manage well than NimH.
Energy densities for top capacity NimH are about the same as LiPo nowadays.
(That was written in 2012. In 2021 LiPo energy densities are now typically somewhat higher).

NimH is a relatively hard battery chemistry to manage well. Charging at low rates is not usually advised and negative voltage deflection under charge or temperature rise are the usual end-of-charge detection methods. In contrast, LiPo is charged at constant-current until a set voltage is reached and then at constant-voltage until current falls to a preset level. LiPo will accept any lower-than-maximum rate of charge if desired, and can be recharged from any state-of-charge with no special conditions. (Handling very low voltage cells is slightly more complex, but all sensible charger ICs handle this - and very low voltage should never be allowed to happen.)

The ONLY reason I would think of using NimH in your context is safety - and if it was my son, I'd consider that I could make LiPo safe enough for him to use. LiPo can "melt down" very enthusiastically with flame, BUT it is extremely rare in practice and taking quite usual precautions should allow a safe result. I would have no personal concerns over LiPo safety in a competently engineered system.

HOWEVER, NEVER use unprotected LiPo cells if you care about safety. The in-battery protection IC DOES NOT serve the same roles as the charger ICs do. The in-battery ones are just to stop people from doing stupidly dangerous things to the battery. That said, IF your charger is properly implemented, and if there is no chance of short or fire potential then most of the protection circuitry is not needed. I say "most" because, if there is e.g. a catastrophic equipment failure and e.g. a short circuit occurs, the in-cell circuitry will usually open-circuit the cell and prevent a fire.

Using the proper charger ICs should allow a very safe and reliable charger to be implemented.

You do not need gas gauging per se - just low voltage cut-out. If you can stop operation at say 3V / cell, that should be enough.

Protected cells should not cost vastly more. If they do, it MAY indicate that the cheap ones are bad ones. You can get utter junk LiIon batteries (and you'd hope to get a price advantage when buying junk :-) - if you were silly enough to buy them. There are enough reputable brand cells around that buying them probably does not cost vastly more. Ensuring that the cells are genuine is another matter. As a working position I suggest you start by assuming that anything bought from a low cost Chinese supplier is fake or out of spec and THEN try and prove otherwise. (NB: Racism? - definitely not!. It's based on experience - many visits to China and time in factories, etc. China is very, very large and has a vast range of sellers in a very competitive market place. In a casual sale, expect a certain portion of the sellers to be 'dodgy' at best.)

Added:

I was going to come back and mention LiFePO4 - AndreKr beat me to it.

Compared to LiPo, LiFePO4 (Lithium Ferro Phosphate) are safer, longer life and have lower energy density. You can buy RCR123A LiFePO4 batteries with 450 mAh x 3.2V capacity. (Some claim up to about 700 mAh but are suspect.) Tenergy LiFePO4 RC123A are widely advertised on ebay and should be good. Tenergy are AFAIK a "rebadger" BUT seem to sell good product. LiFePO4 MUST be charged properly, but are as easy as LiPo to manage. A very simple charger can be built using a constant-current regulator followed by a 3.6V constant-voltage regulator. This setup charges at constant current until Vlimit is reached, and then at constant V. Setting to 3.5V is better.

Here is a randomly found seller of Tenergy LiFePO4 RCR123A batteries. They also sell chargers. NOTE:
Do NOT use Lithium Ion RC123 (3.6V nominal).
Do not use 3.0V Lithium Primary RC123.

The terms RC123, RC123A, RCR123, RCR123A etc are used somewhat interchangeably by sellers. Just be sure of what you are getting.