I have fallen out with my hairdresser...

[bearhug]

Will a different hairdresser know what colours to use to touch up my roots? Would my old salon pass on this information if asked?

The old hairdresser won't be likely to pass on a clients info. It's highly unlikely and if you've fallen out, can you trust they will give you the correct formula?
If you go to a new salon ask for a consultation first, any reputable hairsresser will do this before any colour service along with a skin test. At the consultation you will be able to look at a colour chart and show him/her a couloir similar to what you usually have and they will be able mix up "their" version of the formula.

Good luck!

Well depends if you play it the right way.

If you suggest for example you have moved away temporarily but will be coming back, you may have some luck.