How much for LBS to replace drivetrain and bottom bracket?

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Just ask how much all the repairs are going to be then decide if its worth it to you?
If you are not going to use it much even if repaired I'd just give it away to anyone who wanted it or a bike recycling place.
I think it will be in the region of £150-£200 going on work I've had done before.
 
If the OP is not mechanically minded (seems the case) and bike is worth salvaging, then one of those places run by volunteers who recommission donated bikes would be a good idea. The one in my hometown back there is called the Reading Bike Kitchen (RBK), and they also run classes on how to maintain and repair bikes that are worthy of a second life. Many similar exist throughout the UK as far as I am aware.
 
Changing the rear cluster can be done in 5 minutes after the wheel is off. Putting new bearings in the BB cant be more than 15 minutes. Then there is the price of what ever parts are replace.
 
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