well, it's a tough one since they're mature cats and not kittens. however. you're moving into an apartment that neither cat has been in, which gives you an advantage. here's what i'd do:
your cats must enter the place at the same time, one in your arms, one in your gf's. pick a room that is quiet and not too elaborate, but has places for them to go. put in the cats, you and your gf, and dirty laundry with cat hair and scent on it. make sure your dirty clothes are near your gf, and vise versa, so the cats see that the other animal smell on you is okay. sit 3-4 feet apart. have a dish with a little food in between you and your gf. also have water and a litterbox in the room. and wait. after a while, like an hour, start taking shifts being in the room. when its time to go to bed, use your judgment and assess how the cats have been so far, and ask us or decide yourself if they should be crated side by side overnight. if you thing they're still pretty tense, crate them overnight so there's no fights. make sure you put dirty laundry from the OTHER cat owner in the crate, so they get used to the smells. first thing when you wake up let them out to pee and drink, see how they are feeling after sleeping beside each other. get your coffee and sit and chill with a magazine or book, in the room with the cats. take shifts all day. give times as well for them to be alone, but sit in the hall outside the door to listen for fights. come night time, repeat previous night's rules. you may see them relax in a few days. you want them to really get to know each other before you let them run of the new place. otherwise they can just hide and never bond. but cats can sure bond, they just need reminding! all this can theoretically be done in your bedroom, where you might as well spend a good chunk of time, it's the comfy-est place! and you can watch them at night, which means you could not worry about crating them, if you feel risk-eh.
so, this is exactly what I would personally do if it were my situation. i gave it to you exactly as it played out in my head lol, but you can decide what you think will work best since you know your cats well