The same applies to BrE. We don't pronounce the final "e". The word sounds as if it finishes with a "v". However, the first part of the word clearly contains an "e" as part of "ser". As Mike said, it's not possible to say "srv".
The only difference, for me, is that in AmE you will hear the "r" and in most regional BrE accents, you won't.
Click
HERE for several pronunciations of "serve" on Forvo. There are 2 AmE pronunciations and 1 Canadian English, and in both of them you can clearly hear the "r" sound. If you then play the BrE pronunciation (by me!) you will not hear the "r" sound.