There would most likely be a group effort to keep both Kirk and Janeway out of trouble, of course coordinated by Spock and Chakotay because keeping them out of trouble is a full time job.
Tuvok, Vorik and Spock would agree that while humans are illogical they make for very good and interesting friends.
Paris would be like a kid a Christmas because he really wants to fly the Enterprise and vice versa for Sulu. The pair of them spend way too much time comparing notes on flying starships.
Kim and Chekov probably commiserate about being the forever Ensign. (Depending on if they meet during the TOS series or the movies)
I can see Uhura,Chapel, Rand and Kes being best friends almost instantly.
Scotty would love Voyager’s tech and be super happy talking to Torres about the advancement in technology and agree that the ship is technically theirs the captain just borrows it.
Torres would probably be a little wary about the TOS crew because Klingons and the Federation enemies back then and she is wary of her reception as a Half-Klingon. But Kirk assures her that there is no feelings of hostility on their part.
Bones and The Doctor would spend a lot of time commiserating over how difficult their Captain’s are as patients. Bones would be in awe of the Doctor since he is a hologram and one of the best physicians he has met and Voyager has some pretty advanced medical technology. The Doctor would have a little bit of hero worship for Bones and they would get along swimmingly. Somehow Janeway ends up in sickbay after a brush with death and both Bones and the Doctor have to practically kick Chakotay out of sickbay.
Kirk might try to charm Seven but instantly get rejected. The Voyager crew thinks it’s hilarious.
Kirk tries his charismatic and charming routine with Janeway but he spies Paris subtly shaking his head and then realises that whilst he has been on the charm offensive, the Voyager crew have all been watching him and planning to find a way for Chakotay to cut in and then he realises that Voyager’s crew are practically family and incredibly loyal to each other.
Janeway and Kirk would clash at first with them being Captain’s from different eras, but eventually they work it out and prove to be a very formidable team and also their knack for getting into trouble doubles. They both also clock on to the fact that they each have romantic feelings for their Number One but neither says anything.
The Voyager and Enterprise crews conspire to get Kirk and Janeway admit that they are in love with their Number One.