Outstanding Pearson Learner Awards

I don't see why private candidates wouldn't be eligible, they take the same exams in the same way. However, quite a few students get full UMS in Maths and I imagine any prizes are decided on who gets the top raw marks. (For my experience in the UK for the exam boards AQA and OCR (Edexcel don't do any), any letters telling candidates that they've come top come out on Results Day.

I imagine getting a prize would be a nice thing to add to an application but I'm not sure how much difference it would make.

Any idea how I can get recognised?

Too late. Either you achieved enough raw marks to get an award or you didn't.

If I did, will I be notified by the centre from which I took the exam or what? How can I check if I did or not?

I have no idea as these awards don't exist in the UK. I've told you all I know - other boards tell the centre on Results Day and get them to tell the candidate.