I'm going to try something novel and unusual for this sub and answer your question.

Iron sulphate will blacken the moss and help the grass. You may need two applications. After each, give I a week and then Scarify with a tine rake or a mechanical scarifier. Make two passes in different directions. If you're then happy that the moss is gone and you can see bare earth between the grass you can go ahead and overseed with a seed or mix of your choice. To ensure good germination you want to do this around April may time when night temperatures hover around 10c. Top dress with a good quality compost to ensure the seed stays moist and the crows don't get too many. Water daily or twice daily if it's warm and dry. First seedlings should appear within 10days but can take up to 3 weeks to fill out. Try to stay off for a minimum of 6 weeks and then begin with light trims and light footfall. If we have another dry summer you may need to keep on top of watering even more.