As said broadcasting figures (BARB) are done by member households, in which case whatever device you're using is counted, iirc you use a remote that tells a little box what channel you're watching and logs people in/out.
However TIVO and Sky+ figures, along with on demand can be collected by the service providers for their own use (on demand certainly, and IIRC for both Tivo on VM and Sky+ boxes it would be trivial to collect anonymous data on viewing habits for Sky/VM).
UK viewing figures won't make much if any difference to a US series getting renewed, it might affect how much a UK broadcaster is willing to pay for the rights, but that's about it.