You can definitely use a normal bath, assuming it's comfortable for you. It's true you may not be able to actually deliver underwater because it isn't deep enough to submerge yourself in, but you can still labour in it and stand up or get out to have your baby. Our local hospital only has one room with a pool and it's actually free most of the time. They said it's rare they have someone request to use it but it's already occupied. My friend (at a different hospital) had her baby in the bath.
Personally, I preferred using TENS to the birth pool and really only got in for about 30 minutes before I got out again (had a home birth, so obviously wasn't an issue having a queue for the pool). I don't think it's fair to hog the pool if you aren't going to use it in the early stages, but the best approach might just be to stay at home using your TENS until you feel you need to think about using the pool and then head in so you can get in when you need it. But yes, you can definitely use a regular bath (or shower) if it's comfortable. I actually wish I'd thought to get in the shower (unfortunately, we'd probably already used all the hot water to fill up the pool).