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You'd need a latch lock, keyed both sides. Something like this.
https://www.amazon.com/Double-Keyed-Gate-Latch-Lock/dp/B07HDF9R4R
However to find one that you could directly swap for that deadbolt could be difficult. You may well need a new lockcase fabricated and welded back into the gate.
To make sure the gate is closed after exit I'd put a overhead door closer on it (the things you have on firedoors) as well.
Go and talk to a local locksmith.
Mr_Sworld-
Thanks, that’s exactly the type of thing I had in mind though more work than I wanted to do. My housemate can weld so will probably have a go at it ourselves if it comes to it.
I was considering a keypad entry also, but they all seem to have a latch on the back. Is there any with double sided keypad, or keypad one side and a key on the other?
lmamammal
Does a type of lock exist that will always lock when when closed and then require key again to open? Looking to replace this one on the security gate on the front door, as a lot of housemates are too lazy to keep it locked.