hey, first off I'm not sure of your age. If you are not a legal adult there may be other rules (ie tell parents). If you are legally an adult that doesn't apply.

I'm from the US and I'm not a therapist. From my experience, it depends on the therapist. If you mention a thought with clear action and intent well... they want to save you for killing youself. So actions they might take are ordering you to check in to a psych ward of some kind, somewhere that you HAVE to be there so you can't kill yourself and can get 24 hour care. If it's less urgent/you're not planning to leave therapy and just go kill yourself sometimes a therapist will see you more often, and suggest daily partial programs (not inpatient, but being an outpatient and getting further care). I've even heard of therapist who have you sign a promise saying you won't do anything. It really depends on the GP/therapist.

From what you're describing, you have ideas but 0 actual intent to commit suicides. That's suicidal ideation/suicidal thoughts, which is something that is common to work with in therapy (I think). So are you seeing a therapist? Your GP might send you to a counselor etc to try to help you, might suggest medication.

As long as you're in no danger to yourself, generally it's not a problem.

Also, I'm not sure if a GP has confidentiality, but for sure therapists do. They can't tell your friends and family if you have suicidal ideation - if you are actively suicidal (which I don't think you are) they would probably find out becuase you'd be in a hospital...

hope this helps some. hopefully someone from UK can give some more specific UK answers for you