Yes - it certainly looks like Prunus avium, the wild sweet cherry, also called gean and mazzard.
I say that based on bark, leaves, fruit, habit.
But whether it is the plain species or a cultivar, one can't say. The flowers would be the key, but there are no flowers in the photos of course.
Also, there are other species of wild cherry in the UK, and many more further afield. And in a garden, one might find one from the other side of the planet, eg. Japan or China.
If you look at the trunk, there may be a fairly sudden change in diameter, either near soil level or just below where the branches start. If so, that indicates a graft, which implies the top of the tree is of a particular variety (cultivar) selected for its ornamental and/or culinary qualities.