I am a 6ft 4inch tall driver and need to rent a car in England. In the USA, I use National Car Rental because you can choose any rental car from a host of choices in the car class... this is great for finding cars with good headroom... in the UK I assume I cannot do this and likely will get stuck with a rental car assigned to me at check in that is in the class I paid for... is it easy to switch cars at any of the UK agencies to one that would fit me... dont even mind downgrading in class to get a car with better headroom or leg room.. I do not care about luxury brand, just needs to have the room for example, I would not rent the Mercedes A class or Audi A3 as they have NO headroom...
Like in the States, the clerks seem to have a fair amount of wiggle room in assigning cars depending on what came into the lot when. My 6'2" husband always had good luck with VWs, which should be workhorses in many European rental companies. Even the Golfs he's sat in have decent head room for him
Renting a manual transmission might not be much of an issue switching at the counter. Renting an automatic transmission may be a different story since there are fewer of them. You may get to the court terrible and there's only one automatic on the lot. I always rent from Avis.