I have recently travelled to USA with people who had had covid prior to travelling- the document of recovery is a letter from GP stating you have recovered from covid 19 alongside your positive PCR test result from NHS. Not a lot of doctors will provide this letter, but you can search online for a private consultation who will do it, when we looked it was in region of £60-£90, think we paid £65 per person.
the other option (which my sister took) is if you are testing negative on lateral flows, to get an observed lateral flow through Randox, they accepted this for her.