As you have a sandy soil the reason for the moss must be due to shade.

To have any chance of improving the lawn for the long term you'll need to try to improve the amount of light.If you kill the moss but don't sort out the reason why you have moss(shade) it will just return. Otherwise what you propose sounds reasonable .ie feed it now, weedkiller in spring / scarify and over seed all in spring.

Also there are grasses that are designed to cope with shade better than others. make sure you buy them when you come to re seeding.

Out of interest what percentage is moss and what percentage moss?

I would highly recommend Verdone Extra weed killer for lawns. I find it very effective against a wider range of weeds than granular weedkiller

If the shade is caused by trees can you cut them back or lift the canopy to allow more light?

That is great advice, thank you anchorman. Are you able to recommend a good feed for the lawn for autumn ?

We have removed two large conifers since moving in, and brought down the height of the hedge but not much we can do with having huge trees on the boundaries. We have large pines at the end of the lawn in a wooded area, and large oak and beech and birch down the side. We've removed 3 sycamores as well.

I have already used shady lawn seed in some areas so will overseed with this in the spring.

I usually use "Evergreen" granular weed and feed. but I think they're probably all much the same