Becoming ALT after rejected from JET – Advice?

Throwaway account for reasons.

Hi everyone! I'm aware that threads like these are prolly a dime a dozen, so I'll try and make things quick.

I'm (22M) a final year university student, studying History and Politics. I live in the UK and have been for my whole life, but I don't have any TEFL certifications.

So around October this year, my university held an informational event covering the JET program. I decided to apply, got everything submitted just before the deadline, but unfortunately was rejected.

But it opened my eyes to the whole English teaching industry in Japan, whether through Eikaiwas or dispatch companies like NOVA.

I want to get an ALT position at one of these companies as a way to get my foot in the door and start working in Japan. I don't have any illusions of making it my career, but my plan would be to get a gig teaching English, use it as a base to learn Japanese as best I can for 2-3 years, then try and break into the mainstream Japanese workforce once I get to like N2-1 level.

I'm aware that you should bring some money to Japan – I've already started working and I hope to save £5000 before I make the move.

So I suppose what I'm asking is if this plan is realistic or if I have my head in the sky. And how do I go about applying for ALT positions whilst overseas and not having a degree yet? Should I be looking to apply now, or later in the year?

Thanks a lot for your help!