R.J.
Pools are fine, you just have to make sure they don't get cold (which means short periods of time in the pool). Newborns don't shiver (which is why they have "brown fat" which keeps them warm), so you have to watch their lips and movement. Our family starts swimming with infants as soon as their belly button falls off. But that first year is almost never over 30 minutes in a stretch, and in the beginning (even in heated pools) it's more like 10. What you look for is that they're *active* and splashing about happily, and that their lips are bright pink. Any slowing in activity level, or purpling of their lips (some babies continue to play happily even when getting chilled), and it's time to pull them out. Whenever you pull them out, DO keep them wrapped up and then skin to skin as soon as you warm up, because your baby will set their internal temp to match yours.
HOT TUBS, however are a completely different matter. Infants *cannot* regulate their internal temperature. Taking an infant into a hottub can spike a "fever" in them that matches the tub temp in under 5 minutes. I've seen babies in the ER with temps of 109 who were fully brain dead. Signs above hottubs *always* have warnings about not bringing infants and small children in (small children will also spike fevers to match the water temp, but it typically takes a little longer, like 10-15 minutes)... but I swear, I've had to talk to more parents while roadtripping (gently showing them the sign and explaining why) about hottubs... that I've started carrying a thermometer in my pool tote so parents don't have to run up to their rooms in a panic, but can check them right there. Each and every single time a baby has been temped last spring while kiddo and I were traveling, they've been between 103 &107.