(((ETA, sorry, I didn't actually answer your question or listen to the "at present we don't have anywhere we could have free swim time"...sorry 'bout that...)))
Every kid is different, every parent (and their schedule) is different too. That said, I'll tell you what I do.
I give my kids swim lessons. My first goal was to teach my children,
1. Not to be afraid in (and under) water
2. That swimming is FUN
I have seen my children, (especially my niece who has widespread trust issues), gain a LOT of confidence from this process. She was initially more nervous/tense/afraid/whiny and has ended up being a little shark!
We started by just playing...a lot. At the time, we were visiting my Dad for a month where he lives (on Maui) so we had access to a free pool and a great small waves beach. I was able to take them to the (warm) water every day. At first, we splashed around, chased toys, ran in waves, jumped in waves, followed fish, met other kids, a generally appreciated the water.
Then, after a week of play, I started dunking them. We did "shoot to the moon" where I threw them up in the air catching them farther down in the water until I had them to their chest. Then I'd yell, "Shoot. To...The...MOOOOOOOON!!!!" Toss them up and catch them to dunk 'em under. Another game we'd play was "Save the dinosaur", where I'd toss a sinking toy dinosaur across an imaginary or real line, and we'd dive under the lane divider or the "wall", to rescue the toy from the bottom. I'd have one on my back and carry the other and we'd just run around in the pool. I'd say, "one, two, three" and then "trip" so that we all got our heads wet.
At first (first few weeks) the kids spluttered and didn't like going under. I'd pull back, and we'd just have fun. Once they were comfortable, we'd try again. It wasn't something I ever made them do, or forced - because I didn't want them to develop an aversion. It was a game, and every time they'd come up I would make a HUGE deal of it, "WOW, you are such a dolphin, you are so brave! What an AMAZING UNDER!!! WOW WOW WOW!"
Now, my little niece asks to go under. She'll hold onto my back, and I'll swim under the water with her hanging on. We go under for long periods. My daughter will jump off the edge into my arms and go under. Neither is swimming by themselves yet, and that's okay. We'll practice strokes when they're comfortable going under without being held or remaining physically connected to me.
Personally, I love the water and love the excuse to go to the pool. We have put our going to the pool on pause, because our membership to the YMCA is on pause. We just take it slow and make it a fun trust building/skill building activity. I've got to renew our membership...
Good luck!