Sounds like a couple basements I knew, including ours.

Won't bother the powder-activated nails, but the tiles continuing to stick are less certain. Probably be fine, but I'd score a line where the wall will go and test-scrape behind it to be sure. (If you test-scape without scoring, it'll end up being a project of scraping the entire floor (been there, done that) and you will find some tiles to come up much more easily than others (use a good, long floor scraper, if you go that route).

BTW, those tiles don't help in cutting down the cold at all. I'm curious how you're planning to heat. We installed baseboards (hot water on a separate zone from upstairs, which is radiator heat. It was a bit of a pain, but well worth it. (Tapping off furnace heat is supposedly easier, if you have one.)