When I flush the blow fills up right to the brim (quite quickly) with water and then slowly drains without flushing. Every 3rd flush it doesn't drain at all and I have to get to get the plunger to clear it.

Plumber came last week and cleaned the drain etc, put in new value. Said it was working. But I still have exactly the same problem as before!

Please help!

What valve did he replace and what did he say was wrong with it? I know of no valve that would cause this issue, you have a clogged drain line and he may have pushed it further down stream. Do you recall how he rodded the toilet drain, did he lift the toilet or what?

He didn't say what was wrong. He replaced the flapper and the lift chain. He used an auger, he did not lift the toilet.

OK, a toilet auger reaches a few fet at best. I don't know why he would replace the flapper, it could have been bad but was not the cause of the poor flush you describe. SSince you aren't doing this work and used a plumber call a few others. Ask them to give you a bid or a desription of the work involved. Go with the ones who say the toilet must come up or that they will be rodding from the rooof vent, at least they have a plan

It takes two or three pushes down on the lever for the toilet to flush

When it does flush does it flush normally? If so it sounds like your linkage needs adjusting.
With the flapper seated the linkage wants to have 1/4"of play. Too much play and you get a flapper that doesn't raise high enough. Too little play and the flush valve will suck the flapper down and you'll have to trip the lever again, Good luck, Tom