Cherry Tree and Plum Tree growing in close proximity?

[PrideRavyn]

We have what we think is an ornamental cherry in our front garden (it came with the house) which is a lovely mature tree. Underneath it is a 'twiggy tree', which I have ignored as we don't do a lot in the front garden and it wasn't harming anyone. But last summer it shocked me by producing 3 very tasty plums, and so, the 'twig' is actually a plum tree. Now I pay more attention to it, it's actually quite a reasonable size (a few ft high and a few feet wide, well, its twiggies are a few feet anyway!) and it's nicely blossoming at the moment, but it's growing right under the Cherry, which in comparison is a great hulking beast.

Is it safe to leave it to grow there or should it be transplanted - is there even a way to transplant it or will it be too rooted up now? What I don't want to do is try and move it and end up killing both of them

Funnily enough it's taken me all this time (a year) to even comprehend that it must have grown off a plum stone that someone's chucked down there a few years ago...the romantic in me really wants to keep it happy!