Upgrade a cordless screwdriver (16.8 V) to Li-Poly
You can do it. The fully charged Li-Ion battery is 4.2V so, in my opinion, even 4 batteries pack will do the job. And it is not downgrade actually, because you get higher capacity with lower weight.

But if you want to use 5 batteries pack - do it. Even 21V will not harm your drill. 

The only problem with this upgrade is with the recharging. Usually, the NiMH batteries have external charger, that charges them with constant current. 

Li-Ion batteries on the other hand use much more complex charging algorithm and are very prone to over-discharge and over-charge. 

So, you will need special charger. You can build one by yourself, using the available chips that implements the whole charging algorithm. 

Easier solution is to design your battery holder for standard type of batteries, like 18650 or 16340 (smaller) (it depends of what capacity you want to have), replace them when discharged and charge in the stand-alone charger for this type of cells. 

These cells are produced with internal protection (for example this type) so it is hard to discharge it too much. 

Here is an example for suitable charger.