I don't know when they left England but presumably a good while,(maybe a month ??), before they left Kathmandu for Mount Everest on 10 March. They reached Thyangboche on 26th and started altitude training and then setting up camps.
The first assault on the summit, on 26th and 27th May failed and they tried again with Hillary and Tenzing.
They reached the summit at 11:30 am on 29 May 1953, climbing the South Col route. With the equipment these blokes had this was a massive achievement. They were seriously tough.
The Rolex invoice is dated 20th May which seems a bit out of sync with the dates of the actual expedition. As I wrote before I'm pretty sure that Rolex would have made enormous publicity out of Hillary actually wearing one on the climb. However, if you read the advertising brochure shown it does not claim to that their watch had been worn on the 1st climb, though the photo certainly implies it. Clever marketing
I'm certain that Rolex would have sued Smiths after they brought out their EVEREST model along with claims that their watch was worn on the climb. The Smiths advertising at the time had a quote from Hillary "I carried your watch to the top of the world and it functioned perfectly" and from Hunt "we are more than pleased with the performance of our watches". There's nothing like that in the Rolex advert.
Of course Hillary may have had it in his pocket but it was there. Tensing, as I wrote before, did have a Rolex and may have worn it though that's also doubtful. Those watches were not as tough as their later equivalents and the life Tensing lived was a lot harder on a watch than anything we'll probably know so that too may have not made it up there. Who knows ??