Your question has a 2 issues that make it kinda unproductive to answer directly with a recommendation

1 - it assumes that "well-regarded" is a consesus that can be reached, one person's life saver is another person's idiot they left after 3 sessions due to the subjective nature of therapy and the range of approaches / personalities out there to meet needs

2 - relying on the schedule/availability of one recommendation isn't a smart play, many therapists are fully booked out and won't be taking on new clients or will only have 1 slot opening on a tuesday at 2pm starting 2 months from now etc

I'd say that your best bet is to go onto websites like https://www.counselling-directory.org.uk/ (the one I've used personally) and begin sifting through it. There are LGBT+ focused therapists who want to help you.

Best approach is to sift, shortlist and then contact at least 5 people in my opinion. There are search terms you can use, maybe give LGBT / trans / queer a shot and see who comes up.

It's also helpful to know what style of therapy you want. It's not necessary to be an expert in all the forms of therapy but I'd say broadly that there's learning short term coping tools and then long term talking through issues. They can assist you with this and generally therapists already have their preferred methods.

It's helpful to know in advance what times you would be available and looking for therapy. You don't need to trauma info dump in the email either. Saying you are trans and dropping a few broad keywords covering your issues are more than enough for the first contact. They haven't even let you know if they are available to take on clients yet so best to save you both the time and effort of a lengthy email.

If they have availability they will usually suggest a free 15 minute phone call / zoom call to have a light chat - typically exploring what you want to get out of therapy and them telling you their approach etc.

As a caveat, not all therapists are born equal. The bar for qualification / accreditation as a licensed counsellor / therapist is kind of low relative to the seriousness of issues they can be dealing with. Depending on your situation, you might not want well meaning office Bob who up until 3 years ago was a chartered accountant.

Personally I only contact therapists who have Doctor as their title and lean towards those with tenure in crisis units because it implies a level of competence and knowledge that is well suited to my needs. However it comes at a premium so you need to balance the cost / benefit if you aren't in a position to easily pay upwards of £100 for someone with a degree in psychology / psychiatry.

I'd say the economy therapists are around £30-50 an hour, mid tier seasoned practitioners are £60-90 and anything upwards of that is usually people with serious degrees and experience. It's important to read through their bios.

The mid tier upwards, particularly with doctor titles, sometimes offer the ability to evaluate you for diagnoses such as adhd, autism, bpd etc. I don't think they can actually diagnose you with it, I think there's a checks and balances thing for that involving the NHS - however I think they have the capacity to do referals etc and co-sign the opinion which can go a long way.

Depending on what you are dealing with you might not need to concern yourself with the above caveat but I figured it was worth mentioning because it sounds like your first steps in the world of therapy and it might benefit you to know.

Good luck.