Yes they will definitely need to be neutered, I'm afraid she is incorrect. Male bunnies will almost always fight if they are kept together once they reach sexual maturity, and they usually go straight for the testicles. Something like a change in home is likely to be a trigger, because there is a new territory for them to fight over, so please do keep a very close eye on them. At their age, they should be ready to be done and personally I would be on the phone to your vet first thing Monday morning asking to book them in ASAP. I don't mean to frighten you, but at their age it could happen at any time and especially now they have moved. If they do fight seriously, it's unlikely that you'd be able to rebound them, so I would definitely get them booked in ASAP.
A few months back via a thread on RU, I ended up temporarily looking after two really sweet, very young brothers who I took straight to my vets on collecting them because one of them had partially castrated the other, and he had to have an emergency op there and then to 'finish the job'. It is extremely common, unfortunately.
Hopefully your two are nice and content together and will stay that way until you're able to get them booked in. If they can be done this week, that would be ideal and they can stay together throughout the op and when they come home.