I am a psychotherapist registered with the UKCP. I haven't committed sexual offences, but I do have a prison record for GBH sec20. I can't speak for all colleges but it wasn't an issue with mine. There was however a lengthy assessment for all potential students. My conviction wasn't spent when I did my training.
It is worth thinking that you haven't even received your conviction yet, so it is early days yet. Where I trained we were required to have been in therapy for, at minimum, a year before applying to train, and then required to be in weekly therapy throughout the course. I had been in therapy about 4 years before applying. I continued my own therapy for 13 years. It is important that you work very deeply with your own issues before you even think of putting yourself in the position of therapist. That would be the case even if you had no history of offending.
Integrity is VERY important for therapists and I can't imagine you going through a full training without you talking honestly about this issue, your history, fears, life problems etc., so check out with any potential college at interview, otherwise you will only get into problems later on and /or fail to be properly entering into the training (which is always deeply personal).
A fair percentage of therapists in training want to train because they have had therapy due to a difficult past. You will likely encounter fellow students who have been victims of sexual assaults etc., so expect that. Also, some therapists in training may have had convictions, drug problems etc. etc.
Many of the older established training institutes are private colleges and will have their own views on the matter, and their own criteria. e.g. even your age can be an issue for many institutes! I guess University based trainings will serve much larger bodies of students and so might be more risk averse.Placements
Placements may be an issue as they will likely want a DBS check, and if there is a requirement for a psychiatric placement (mandatory for UKCP) then most psychiatrists work within the NHS and will be very risk averse about working with you.
You might possibly find more success with some specialist units that deal with treating offenders eg. the Caswell Clinic. I ended up doing my placement with a drug rehab centre (the director was also an ex. con!) Organisations that work with people who have offended in some way will be less likely to 'pigeon hole' you... but you must be willing to expect some disappointments.
I suggest you contact https://www.stopso.org.uk/
this is an organisation for therapists that work with sexual offenders or those who are worried they might offend. They will likely know more about the law in this area and will likely know people who may be sympathetic about what you wish to do.
Don't give up!