If everything is fine you can go home just about as soon as you've got cleaned up! I was home 4 hours after dd2 was born and would have been the same with dd3 except that I lost a fair bit of blood so they wanted me to stay overnight - it was 5 in the morning! We compromised and I stayed till about 3pm so 12 hours after birth.
I was told 6 hours (as that is the time they estimated every check would have been done).
I had a "normal" VB on just G&A so no drugs to wear off etc.
I chose to stay overnight thinking it would give DH a chance to get some sleep - and at 2am regretted it completely! DH could not sleep as he just wanted to be with us so we all ended up in a tearful mess and sleep deprived by the time we did get home the next Morning.
A senior MW will come and visit you 12 hours after the birth to do the paeds checks after a homebirth.
As far as I know its 6 hours after delivery, but it will also depend on the time of day and when there is someone in the hospital to do the checks.
I have to say, IME I have had far better care for my 3 hbs than my 1 hospital birth.
My only word of advice is if you go home soon after, or have a homebirth, stay in bed for 24 hours minimum. You might feel like Wonder Woman, but you won't get those first hours back again. You should be being cared for, not doing the housework
In NI its 6 hours, make sure you let them know your intentions so if everything goes well they will be on the ball, although with dc3 dh still had to go in day after I was discharged to get vit K as it wasn't ready at time, wouldn't have minded but I had to stay an extra night as they were to busy to discharge me the day before