Replacement timing: bottom bracket after gear and chain
There are good reasons to avoid using a very worn chain for to long as it can wear the cassette requiring both to be replaced. 

I doubt that a small amount of play in the bottom bracket would move the chain enough to cause unusual wear to the chain, rings or sprockets. There may be more chance of a problem on a single speed or fixed gear bike where the chain has less give.

There's a small risk of poor shifting with a wobbly bottom bracket, so perhaps take care and avoid any gears which rub.

Otherwise, as long as the shop replaces like for like, you shouldn't have a problem. Likewise, worn gears should not adversely affect your bottom bracket.