I saw a college therapist who refered me to my GP years ago for antidepressants. I didn't say outright I had suicidal thoughts but then, the questionnaire they get you to fill in tends to reveal all that. They didn't pick up on it though- oddly.
Both were pretty good in their way. The GP was very sympathetic when I just burst into tears. The meds didn't do anything for me and I found the therapy intrusive. (I guess it has to be!) Workwise at that point, things improved for me- which was always my best coping mechanism. So, I quit therapy and didn't renew the meds. I think it's worth trying though- if you have any desire to.
As to who you should contact, as Dot said- it depends how badly you feel really. If you want immediete intervention- then a Crisis line may be the way to go. But I suspect they might initiate a welfare check if you tell them you are suicidal and have a method to hand.
I guess you need to make it sound serious enough- that they get you support quickly but, not so bad you get sectioned- I imagine.
Isn't there that app: Betterhealth to put UK residents in touch with therapists? I don't know how good it is though. The member @FireFox didn't seem to have much luck. Whatever you do though- I think you need to pester in the UK. Our health service is so overburdened. I don't think they have the time to check up on people now. I think you have to push for stuff and, keep pushing!
Good luck though. I hope you find something that helps.