As you pointed out, for reasons of confidentiality, work experience isn’t generally a way to make entry into psychology. I don’t know the regulations in the UK. In America, becoming a psychologist takes years of schooling (at least two graduate degrees) as well as years of supervision. Teaching also requires a university degree.
However, any volunteer work you do that involves helping people will help you get ready for either profession. To become a good psychologist requires more than schooling. It also requires developing familiarity with and compassion for many kinds of people who are struggling with many, many different issues. The more you broaden your experience with people from different walks of life, the better. You can prepare yourself to be a teacher by doing volunteer work with children.
I am impressed that you are thinking about this now. Too many people go into these fields on the basis of an idea, not experience. Volunteering at a few places will tell you if you have the talent and patience for working with people who are struggling in some way. If not, you may find something else that appeals to you.
I wish you well.