Becoming a tutor

To start with, avoid using words like 'skillset'. A. it's not actually a word, and B. it makes you sound a bit of a díck.

What experience do you have in administering diagnostic tests to ascertain gaps in learning and key skills which need working on, and then delivering a bespoke teaching plan to fill those gaps? How about examination and marking? What about coaching? Have you dealt with difficult and anxious parents before, and how would you assure them that you were worth their hard-earned cash?

Yes, although the age group was a little lower, I have a fair amount of experience with assessing children and working with anxious parents. I used to be a helper at a children's club and worked with them to develop cognitive skills and help them with tasks in 'fun activities'.
I've never specifically done tutoring, which is much more in-depth. This is why I'm charging a little lower to some of the prices I've seen around online. Do you think it's still too high?
If you want a proper answer on to how I would assure parents, I would tell them that I was at the top of my class and am attending a top university in the subject. I also have much practice in marking answers and identifying areas for improvements as we often did peer-assessments in class and I frequently reviewed my own work when using practice exam papers.

I think this all sounds great

As for the price, I think you're actually selling yourself a bit short - £20 per hour is quite cheap really. Tutoring is all done by word-of-mouth essentially, and it's all about parents talking to parents. If you want to do it independently, the easiest way usually is to tutor the child of someone you know (or your parents know), do it superbly and then get them to spread the word.

I think you could charge £25 per hour, particularly if you're doing an exam class (i.e. GCSE)

I've asked a few Psychology Year 12 students I know who said they'll get back to me but I'm not sure if anything will come of it. I might put up a few ads online or in person to get a head start.
Thank you for your help