Any saving in flying to the US, living in motels, eating all meals 'out' and using taxis to get around may be less than you might imagine. It does depend on how many hours you want to build and how quickly, but remember you also need to prepare for the flight tests and be thoroughly familiar with UK procedures, nav and met. You can lease blocks of hours - usually dry - on light aircraft surprisingly cheaply here. Finding a local airfield/airstrip to base it at while you fly off the hours and live at home may be almost as cheap. You can often dodge landing fees en-route by researching where they waive them if you uplift fuel too. You'll learn next to nothing useful for the tests in USA ("hours are hours" is not a helpful or accurate suggestion imho) as procedures and style of flying is so different and you will certainly pick up habits frowned upon here which you'll need to lose before coming in front of CAAFU so a substantial amount of time will need to be allocated for preparatipn for the tests over here.
Personally I think that unless you need to build hundreds of hours you're possibly better off overall doing it at home and/or touring Europe.
There's a lot of research to do!