At any one time today at EGBB, I had 10-15 pilots on my frequency asking for vectors. Of those, 90 percent would be asking for vectors to Manchester or Heathrow, though I was Approach for Birmingham.
Just because you see an open frequency doesn’t mean you have to interact with it. In fact, if you’re at FL320 (although why you would be that high for a flight between EGCC and EGLL completely eludes me), you’re not in the airspace, even if it were for a field involved in your travel.
Interacting just for the sake of it clutters up the frequency and takes time away from the pilots who are actually meant to be served by the frequency.
You do not need to ask Birmingham Approach for Flight Following to EGLL as you cruise overhead at FL330. It serves no earthly purpose. You certainly shouldn’t be asking for an ILS approach to a specific runway.
If you want Approach service, try the Approach frequency for your destination, not just any frequency you happen upon along the way. And you most definitely do not need to switch to tower and ask for transition, or use “going around” in the completely wrong context.