Tree has split

I think you can leave it safely alone. Maybe you should plant a new one in the autumn, though I have had a lot of trouble trying buy one!

What a beautiful tree! And neat garden.
I didn't know damson trees could grow to that size, maybe it had outgrown its strength.
Maybe you should ask an expert, but I would say that if you want to keep the remaining part you should paint the wound with some special sealant. Can't remember the name. Brown sticky stuff.
In this warm damp weather some kind of fungus/mould is likely to invade.

on a recent gardening programme the advice was not to paint it with stuff , that seems to seal in bacteria , not keep it out , I would seek advice from a tree surgeon

I would also say leave it.
I have an acacia tree in my front garden & after some very strong winds, the main trunk snapped. I googled what to do and the answer was nothing!
To my surprise it just carried on growing & the next spring was a mass of blossom.

We are going to leave it. Have got someone to cut it from standing trunk. He has angled the cut so water will run off. Fingers crossed. Thanks for advice.