Autumn is the best time of the year to do it, so I'd suggest a ferrous sulphate treatment (its the active ingredient within more expensive lawn treatments - you can buy a tub from Amazon or ebay for under £10). This will turn the moss black over a few days. Then you need to rake the hell out of it (back-breaking work on a large lawn so the faster option is to hire or buy an electric/motorised scarifier/lawn rake. This will lift all the dead moss and thatch, which you need to lift off and stick in your garden waste/composter. You'll initially be scared by the sight of your near-bald lawn...
But don't be concerned, mix new lawn seed with topsoil/sand and then spread it over and tamp it down by walking over it. Then daily watering (ideally early evening so it stays moist through the night) and the new seed should start to grow in a week or two.
To make an even better job of it, aerate the soil after your scarifying/raking by spiking holes in it every few inches
Good luck