Will weeping tiles prevent all dampness in my basement?

Weeping tiles probably won’t stop all dampness, since they’re only meant to move liquid water away from a basement, not water vapour. And you’re right to be concerned. Moisture can ruin a finished basement and fill your house with harmful mold spores. I’d recommend a coat of Xypex to stop both liquid water and dampness coming through walls and floor. It’s a powder you mix with water into a slurry, then slosh onto damp concrete with a long-bristle brush. I’ve used it in my basement and it works spectacularly. The only problem is that it can’t be applied over paint. It must be free to soak into the concrete pores.