When I had this issues with blocked drains and wet carpets as a result, I following these steps, in this sequence:
1) Get some old dry towels, laid flat on top of the carpets, to soak up/get rid of most of the surface moisture on the carpets
2) Turned over the carpets, used the flat bladed plastic edge of an ice scraper (gently!) to squeeze out most of the moisture from the foam backing of the carpet: if you're not gentle doing this, then foam will fray, so care is needed
3) Repeat step 2), but on the surface of the carpet - if there is still any significant moisture in the pile
4) Repeat step 1) again
5), Leave carpets in boiler/radiator cupboard to dry out - I did this for about a week.
After this, all was well, no residual damp or staining to the carpets, so all good
