Regarding planning permission, it would depend on where the garage is sited.
If it is within 1 metre from a boundary or can be seen from the road then it will require planning permission.
You or Taylor Wimpey would need to apply for planning which may take several months before a decision is made.
Even if Taylor Wimpey are prepared to do this and build, the garage, it will cost you much more than if you got a local builder to do it. Choose the right local builder and they would probably do a much better job of it too!
My advice would be to look to buy a house that meets all your needs.
Taylor Wimpey may even have a clause in the Title Deeds that prohibit you building anything for a few years too.
You also have to ask yourself (or their sales staff) why Taylor Wimpey didn't want to build the garage and maximise their returns on your plot?
Taylor Wimpey are not the worst housebuilder, but third worse is no endorsement!
So look to buy from a better 'five star' rated housebuilder (Miller or Barratt), or even better, by an older home.