It's all down to your own employers standard terms.
I get paid overtime for any work out of normal hours at a fixed rate regardless of what day the time is worked and if any of the hours are between Midnight and 06.00 my next day start time is delayed by the same number of hours (recovery time), providing the next day is a normal working day.
If it's a Saturday for example I get no recovery time.
If I worked all night till 06.00 I wouldn't be expected to work at all the next day, but would still be paid.
My previous employer would have had me turn up for just the last 1 hour.
I too have worked permanent shifts and could have done with a nightshirt but managed with just a sleeping bag.