Maybe changing both the bottle and the drink at the same time is too much for him?
Don't forget it's the CM's job to look after your child, that's what you're paying her to do, and that includes feeding him, so if it takes that long to give him his bottle she'll have to do it, iyswim. I'm all for making things easier for CMs, but there's a limit.
I did the same, tried to get DS to take a cup before he started nursery, but gave up in the end.
I know the cup thing gets pushed by the HVs a lot, but tbh DS has never drunk milk out of a cup. He's 2.4 now and still has a bottle first thing in the morning - cup for water and juice, but milk comes out of a bottle as far as he's concerned. I figure as he drinks it down in one go, it can't be too bad for his teeth.
Don't know if any of this helps, but if you're really keen on the cup thing, why not try formula in a cup, and then switch to cow's milk once he's used to the cup? Or change the milk first, then go onto a cup?
HTH