Plant identification... Plum, Damson, or something completely different?

We've got a few trees in our new house/garden which look like they've previously had the chop, but of which a lot of tall wispy branches have grown back.  

I'd like to know what they are if anyone can identify?  I'm thinking plum but might be waaayyyyy off the mark!  

I'd also be interested to know whether they will still be productive in this condition or if they've had their day?  

Many, many thanks