Pruning an overgrown lilac.

It's all leggy. I think (!) that if I cut it now, there won't be flowers next year. However, it would be good to tidy it up a bit. If I cut some branches, are these likely to sprout, so there will be some flowers next year and more the year after, or will the truncated branches just die? All the growth at the moment is up at the top. Few if any lower growths bar an abundance of suckers.

 

Cut branches should reshoot. You could cut a third out this year, a third next etc. I have hard pruned a lilac to about a foot above the ground, it survived and flowered again a couple of years later.
I would remove the suckers before they take over.
Some lilacs are grafted; if yours is then remove any growth from below the graft as well as the suckers.