Maybe check on Google if it needs a pollinator if a single tree. There may be none near to you, if that is the case. I have checked and no pollinator required. Occasionally fruit trees have a poor perfoming graft, so think of that too. As it might have been expensive, perhaps you might refere back to the seller and ask for replacement or money back if you have the invoice. It is probably aphids causing this mess on your car. ?spray as no fruit.

Yes, aphids do make a sticky mess

Is the tree blooming in spring? The Greengages that I am familiar with don't need another variety for pollination, but they do need visits by bees to self pollinate. A lack of bees in your neighborhood might be the problem. Another thing to check is what the weather was like in spring--untimely frosts or windstorms could rob you of your crop, too.
If the tree isn't blooming, I would ask how old it is--often fruit trees will grow for 3-5 years without trying to produce fruit, even though they may have had blooms or fruit in the nursery container. A heavy pruing could also set back the clock, since Greengages bloom on wood that is 2-4 years old.

Thank you everyone for your advice. I will be using a combination of your answers. lol. The tree is in need of a prune and I will then give it a spray. Will go to our Garden centre and ask for a suitable aphid spray for fruit trees. I am loathe to cut it down as it has so many memories. It would be lovely to sit under it to shade from the HOT sun when we get it in East Anglia. Thanks everyone

Sorry, Schaca, what I was trying to say--incoherently--was that a heavy pruning would keep the tree from blooming or bearing. So, if you want fruit, don't prune, or be very cautious and conservative about it.

Sorry Tugbrethil, Ok I see what you mean. i won't prune. just spray and hope that sorts it out. Thanks for the help

You're welcome! : )