Don't worry we will usually yell immediately if its something serious.
And asking after we have sorted our poo out somewhere downwind or leveled off on a vector will get you your answer.
I have only had two pilot GA's in the last 5 years and two times had to tell the controller to standby for the details. And for the record one was unable to confirm a landing clearance after getting it at 8 miles and both us not being 100% sure we had one after a ruff approach. And the other one was the electrics for the gear indications going tits up just as we selected gear down.
So I know where the OP is coming from. Its more of a when you ask than if you ask. TWR asking while your still in the guts of flying one isn't the best time.