You shouldn't need to "flush" everything out.
You should be able to isolate the pipework at gate valves / stop valves, likely close to the hot water cylinder, water tank or (if a combi), the boiler.
Once water has stopped flowing at the tap, you can cut the pipe - there will be some water in the pipe, opening the kitchen taps (assuming they are downstairs and lower than the bathroom) will drain the pipes. You can then fit some isolation valves, or some flexi tap tails with isolation valves fitted.
Got any photos of your pipework / airing cupboard / boiler?
Isolation Valve:

Isolation Valve On Flexi Hose/Tail: