Hi,
It is very unlikely you will get psychotherapy on the NHS and certainly not without many months/even years of waiting. There are long waiting lists in the NHS for appointments, then you need assessing and then more waiting lists. Many NHS consultants just treat with medication - there is a shortage of NHS psychotherapy for all people in the UK. I am not sure you would qualify immediatedly for NHS treatment, as a non-citizen so you would have to look into that - just as a UK citizen can't go to the USA and get free treatment. You can see a Gp as a non-citizen, or any dr if you pay in the UK. However, if you have private insurance I would definitely follow that line and see what you can get privately. There is quite a difference in quality of care in mental health between private and NHS.
If you need help urgently when you are in London, the only way to get access to help (and I use the word help in a very loose sense) is to either see a Gp, who you need to be registered with or go to an A+E department and ask to speak to a duty psychiatrist (with generally about 4 hours of waiting). This will not mean you are admitted to hospital, that is quite rare, and you certainly will not jump up waiting lists and will not get any talking therapy in a crisis. You may get short term medication and then told to see your Gp. You will need an address to register for a Gp, although can see one as a temporary resident (say of a hotel) for a certain number of weeks, but then you will need to move GP practice when you rent somewhere.
If you have enduring mental health issues you will need to register with a Gp and then ask to be referred to a psychiatrist, who manages your care in the NHS. I see mine 3-4 times a year, for 15-30 minutes a time. Again you may qualify for this privately. You may then be placed under a CMHT - a community mental health team where you will be allocated a care co-ordinator, either an Occupational therapist, a Social Worker, or a community psychiatric nurse. Then if you spiral down you phone them first - only contactable between 9-5 Mon-Fri and if they think you need it you can get access to a Crisis team. The Crisis Team can support you day or night, but they will only see you generally once a day and it will be a different person each day generally as that is how they work. There will be no continuity, no talking therapy and all staff you see are at nurse level, unless you need a Dr for medical treatment. They can advise you are sectioned and action that, place you in hospital and help you get access to short term medication to help. After that you are referred back to your cmht.