Had the same problem, was told to buy another plum tree, the reason I was given was that they do not like to be alone and will only flower if they have a mate. I thought this was a daft idea but my immediate neighbour bought one and mine did the works! I refuse to comment further.
I also have a greengage which took seven years to produce fruit. I did yell at it in the December 'If you don't bear fruit this year I am chopping you down'. We got two greengages that summer. So, I gave it one more chance and it was full of fruit the next year.
My conclusion is that plum/damson/greengage trees take longer than apple pear and cherry to produce fruit. It is worth waiting for though, the fruit I have had from the supermarket tastes like cardboard by comparison.
Now the problem is using all the fruit with in jams. Crumbles, pies and plain stewed collecting in the freezer. It takes some eating. I have even made plum sauce done the Chinese way.