As an American English speaker, if I hear
You know the drill.
I expect it to mean that I know the routine or the procedure for a given situation. The Free Dictionary gives a similar definition, "To be familiar with what happens or what needs to be done, without having to be told."
Was this used in the British isles as well as in the US? I ran a search in the British National Corpus (BNC), a corpus from the 1980s and 1990s that includes lots of text samples and transcripts. I turned up six results for "know the drill," without any other gloss or explanation. I'll repeat a few of them here:
Different sort of people turn out on the hunting field these days, too. Some of them hardly seem to know one end of a horse from another. No hunt manners. Not their fault, they just don't know the drill. (A season for murder. Granger, Ann.)
' You know the drill?'' Only vaguely. You tell me again what to do.' (The other side of paradise. Barber, Noel.)
' Come in, Russ,' Graham said without looking round.' So, you're off on your travels again. Where to now, or can't you say?'' You know the drill,' Graham said.' You're going after Bernard, aren't you?' (Time of the assassins. MacNeill, Alastair.)
Now, that's six results in the BNC, which has 100 million words. COCA (the Corpus of Contemporary American English) gives 578 results in a corpus of 1 billion words, or 57.8 results per 100 million words. It's possible but not conclusive that there's a discrepancy in frequency of use, but at the very least the BNC bears out that British audiences would know the drill.
Bonus evidence: a UK band called "You Know the Drill."