I'm wondering if their only son, Charles Fletcher, was perhaps the photographer?
I hope someone is researching the Fletchers of Worthing as it would be wonderful to return the pictures to a descendant.
I wondered that too. I suspect the old chap in the car (and other photos) will be Alexander Pearson Fletcher, but my impression was that the driver in the head-on photo of the car was younger. I wondered if that might be Charles - the photo could have been taken by someone else under instruction, though I'm fairly sure that one of the photos I have of a similar age was taken by a keen amateur photographer with some kind of self-timer. Note that the car is stationary in both photos. (I note vintman's earlier comment about the lights, but I still think the question of it having a mirror or not probably suggests two different occasions.)
I still have a few queries in my head about locations. I had a look for Fletchers in and around Carshalton in the 1901 and 1911 censuses, but didn't see anyone obviously related. However, there must have been some reason for them to be there.
Then there's the car with the Norwich registration. Had someone driven down from Norwich, and if so, who? Or was the car second-hand? The Fletchers were clearly wealthy enough to buy new, and something much bigger, though they might of course have preferred not to - Alexander may have been a canny Scot
.
I've also been looking at some of the other photos in the collection. I only know it from pictures, but could QP-06 be Box Hill? Is it the same as
https://www.telegraph.co.uk/travel/activityandadventure/walkingholidays/8356605/Box-Hill-Surrey-walk-of-the-week.html? Might some of the HP-xx ones be there too? And the bridge at HL-29/30 looks identifiable too....