Barrister are the ones who prance around in court
and solicitors prepare case work - write wills and so on
this is now blurred and solicitors can plead in court ( have a right of audience ) but I dont think there are any who have the bit of paper that allows them to appear in the Supreme Court
and they ( solicitors that is ) can become judges
https://www.lawgazette.co.uk/features/how-to-become-a-judge/5049527.article
you sortta have to do the course and get the bit of paper
We need more facts if you can give them
Your question seems to imply that abyone can walk into a solicitors and scream "I appeal!" and some paper shuffler will sidle up and say "Yes sir I will hear your appeal now"
it sort of doesnt work like that