I think the thing to be aware of, as
@KFdancer pointed out, is that things do not happen quickly and especially as you are not part of the system and can't be until you're here, it will take some time. It can also massively be dependent on where you live.
A quick Google found me this for Malvern:
https://www.hacw.nhs.uk/childrensslt/ There is an email address where you may be able to contact someone that might know wait times depending on need.
Again, they aren't really going to be able to help you until you're here. And by being here, you're going to need to register for a GP and a school. My son was referred by his nursery. If I remember correctly he was referred at 4, and was finally seen by the referral when he was 6. We chose to go private because 1) we didn't feel like he could wait but also, and perhaps more importantly (and unfairly) 2) we could afford it. It was very expensive-but we felt like worth it. We entirely acknowledge our privilege in this regard. We were also very lucky as I mentioned before that we could use the services of a local based charity called the Willy Russell Trust. We got in within a few months and a speech and language therapist was able to assess my son's stammer and also the extent to which they felt he might grow out of it/might need specific therapy. Had it been worse, we would have had therapy via them-way earlier than the referral that we had.
Once we were referred, we had to go to the therapist-it wasn't someone that was in school. This could be different in different localities but perhaps different from the USA where they might be within a school or school district (I just got pulled out of class and they were in my school.)
Others on the forum may have more substantial information regarding mental health but this is another area where you may have a wait and where charities and private care may have to fill the gap. My husband had a mental health crisis at the end of November and he's finally getting on a treatment programme starting on Monday. I appreciate there's a gap because of Covid, a greater need for mental health services, the Christmas holiday when things basically shutdown and a lot of back and forth between our GP and psychiatry services, which may be localised problems. But it's been incredibly difficult and making phone calls and demands haven't made our experience any easier.
Think you just need to be aware that continuity of care across the ocean will almost certainly have a break and might not necessarily be what you expect. As said before, you can prepare but there are some variables that you won't know until you are here and see the strain of the NHS services in your own locality and what you may/may not be entitled to.
I'm not saying don't move-I've spent almost my entire adult life here, so this is the only medical system I've used, really. But there are gaps and difficulties (and I love the NHS!)