If you are going to turf the new lawn work can start NOW (early August). Clear the old grass, either kill it off with weed killer (Weedol or Glyphosphate) which will not harm the soil or hire a lawn stripper to take the turf off.
Before ordering the soil have the supplier work out how much you need to raise the lawn to whatever hight you want. Make sure that the new soil is ACID not Chalk for the best results. After the turf has been removed spread the new soil on to the old soil (DO NOT dig or rotavate the surface) as this will make a flat lawn more difficult to achieve. When you have raked the new surface tread by pressing your heels in to the whole surface and then rake again to make it flat as you can. Apply a dressing of Bonemeal to the soil to incourage root growth of the new turf.
Leave the soil for 1 month to settle and then tread and rake again - this can be done twice more leaving 2 weeks each time for it to settle again over the next weeks until the surface is without humps and hollows. By now it will probably be October or November, the ideal time for turf laying - the winter rain prevents the need to expensively water the new turf until spring next year. It is a slow process but the end result should be excellent. One important point is to buy the best turf for your purpose so again, ask advice on turf quality and price. Make sure you do not have Annual Meadow Grass in the turf as it is impossible to get rid of - I know from my own experience!
I have seen all this at my golf club many times and the finish is beautiful. Good luck.