Best Time for a Lawn Reseed?

I know it has not been "lawn" weather for the past few weeks, but I have had it with my front lawn. It has been getting worse and worse over the past years.
I have quotes to kill the lawn, heavy scarify and a reseed process. As we cannot go on holiday this year keeping a reseeded lawn watered is not a problem.
Now is June an OK time to do it or should I wait until September ?
I just want to get it done - fed up of the blooming thing
TIA

 

Cheers John, I understand that grass seed takes 3 weeks to show and 8 weeks before 90% of it germinates. So there will be enough time for it to establish before the weather deteriorates?
I could get them to start the Glyphosating mid August, as it takes two weeks to kill the grass. The reseeding could then be completed early September.