Do expats get a vote in the forthcoming election if registered at the embassy or consulate?

It's not quite as easy as the other poster put it.

If you want to vote, you need register as an overseas voter. You can only do so if you were registered to vote in the UK (i.e. lived there) in the last 15 years. If you left the UK before being eligible to vote, i.e. before the age of 18, you may register now if either of your parents was registered in the UK at that time. The Electoral Commission has a website explaining how to register as an overseas voter.

Yes, they can vote by post or Proxy.

Yes, if the consulate services has you current address you will receive the necessary documents and will be able to vote.

PS : It is assumed that the poser knows the basic rules of consulate registration along with basic knowledge of expats to vote !
That is for G...e !