The best place to go to build connections and becoming the best you can music-wise will always be conservatoires in my opinion. The whole point is that you study whilst actively building a professional career.
Leeds conservatoire is a good option, they've also got some really great facilities there. I don't know much about London College of Music, but it seems good and like it has good facilities, and has options for song writing, which seems to be right for you. Same with Guildhall, don't know much about it particularly, but have heard good things from friends.
Just a few other options for you:
Trinity Laban-BA Music Performance and Industry
Leeds Conservatoire-popular music with production or production with popular music (will give you a more solid musical grounding than just production).
I would honestly say apply to London College of Music and Leeds Conservatoire this year, do everything you need to do, if either seem right for you, go for it, if not, do this:
Have a year off and apply in the next cycle. The conservatoire application deadline has already been, so you'll probably struggle to get a place or even interview/audition at any right now (guildhall and Trinity Laban)
The deadline for conservatoires is normally around the 1st October. Use this year to work on your portfolio and personal statement, and get that application in on time. I'd say your best options are:
-Guildhall (own application through their website)
-Leeds Conservatoire (ucas undergrad)
-Trinity Laban (ucas conservatoires)
-London College of Creative Media (London College of Music, new name) (ucas undergrad).
I'm also assuming here you know these are all undergraduate degrees, so they'll be the same level as your current degree you already hold.