You may want to cross post this over to /r/legaladviceUK to get a better regional response (this sub is mainly US users)

But you are right in your suspicions that you can't apply for criminal injuries compensation without a criminal conviction. You could try to chase things up with the police, but (as you already suspect) without any witnesses they may not be able to proceed. You can make some noise but ultimately you can't force them to do anything.

This doesn't eliminate the possibility of you taking private civil legal action against this person. The standard of proof is lower than for criminal proceedings (in ELI5 terms you just need to prove that the most likely cause for your injuries was this assault, rather than having to prove it beyond a reasonable doubt)

In order to do so you'd need to speak to a local personal injuries lawyer. Most will offer a free initial consultation to discuss your case before deciding if they want to take it on.

One final question though - what are your actual damages (i.e. what would you actually be suing for). You need to be able to demonstrate some sort of financial loss to have a case. It seems like you didn't lose any time from work, and I assume your medical bills were covered by the HSC? Unless you have some sort of long term issues arising from the assault you might not actually have anything you can sue for, and you might be better off just moving on with your life.