Yes, the bamboo I mentioned is hardy to -25C, and is very happy with English summer temps, but does need to be watered in a drought. If you are happy with 5m, then most bamboo suitable for the British climate (there are many but be careful about invasiveness, only Fargesia and Thamnocalamus are reliably non-invasive.

Someone mentioned leylandii and eucalyptus as fast growers. True, but they won't stop growing.

I think if you want a "tree" that stays below 5m, you are really into the category of largish shrubs, such as the photinia someone suggested. There are plenty of others, such as Cotinus, many kinds of Berberis, Dogwoods, Philadephus, etc, etc. A fruit tree on semi-dwarfing stock will also be OK.