C&G vs FAA
If you do the C & G course, it should* entitle you to undertake the shorter FAA flight dispatcher licence course.
*I did the C & G course a long time ago and then approached a well known dispatcher school in Ft Lauderdale with the details of the course and was told that I would be able to take the shorter duration course which is also cheaper. I ended up doing the longer course but for reasons of course schedule. Things may have changed but it is worth a shot as you will save alot of time and money.
If you only have enough funds for one then take the FAA.
The advice given re a start in crewing is good, if you can land a spot as a crewing asst then you'll be off to a decent start.
The only airline, that I know of, in the UK where the FAA licence is mandatory and where you might stand a shot at a starting role is Saudi Arabian Airlines at their dispatch centre in LHR. They only take FAA ticketed people as they operate to FAA 121 regs. Not too many in UK have that licence and when Saudia have some vacancies, most ops bods in UK are not eligible. They are a major airline and a few years there would be good on your CV if you can handle the high cost of living in LHR area but relatively low SV salary.
The good thing is that it is 100% pure flight dispatch, no monkey business or blurring of lines with gnd handling or crewing or any of that other stuff.
My 2c worth.
PM if you'd like more info.
Happy Hunting.