It varies from hospital to hospital. Remember that you're not technically overdue until 40+14 as normal pregnancy covers 37-42 weeks. You also are not obliged to accept induction and can request expectant management instead.
This is the NICE information for women on Induction of Labour so most of the technical/medical info should be in here.
Basically baby's heartbeat will be checked, you'll have an internal and then given either prostaglandin or have your waters broken. Once you've had your waters broken, if nothing happens you'll move to the syntocinon drip.
Induction in the evening isn't recommended and has been shown to make mums unhappier (probably due to the lack of sleep before birth) and increases intervention (again, if you've had no sleep due to the induction process you're more likely to be knackered at the end) - this is from the NICE Clinical guidance, rather than the stuff they give to women.
If they admit you but do nothing and suggest starting you off the following morning, my advice is go home and return to hospital in the morning. If you've undergone no medical procedures (eg no prostaglandin, no ARM) then you have no need to stay in, and getting a decent night's sleep on the antenatal ward is going to be near impossible.
Most antenatal wards don't allow partners to stay out of visiting hours or overnight, so until you're in a delivery room then your birth partner won't be able to stay with you - even if you're in the early stages of labour on the antenatal ward :(