Really depends on how busy the crossing is. They have 4 hours (from 6:30 to10:30) to unload the incoming wave of driven vehicles, then the unaccompanied stuff, before reloading the new batch of trailers only. Finally they load cars and driven freight and this tends to be in the last hour before sailing. Although they ask you to be there at least 2 hours before sailing time, I would expect it to be at least 9.45 before you have a hope of getting on board, but once you are, you should be able to collect your cabin passes and get in there rather quickly.
Birkenhead appears to have a slightly more efficient loading operation in place than Belfast, from my experience.