The oak leaves aren't a problem; they compost down fairly quickly. Go easy on the fertilizer/feed. Too much will burn new seedlings. Ground should be kept barely moist for germination. Too wet will promote fungus and mold. Even "shady" seeds need a fair amount of light.

Lots of good advice here
https://www.lawnuk.com/faq/overseeding-existing-lawn

A good lawn takes time to establish. When mowing never "scalp"the grass, as you discovered. If you cut off all of the grass blades there will be little left for the grass to use for photosynthesis. Better to use a medium height cut more frequently. Getting a bowling green "velvet" finish can't be achieved immediately - it can take years - and it also depends on the type of grass seed you sow or the quality of turf you use. Choice of seed also depends on the kind of lawn, its intended use and its aspect. A rye grass mix intended for walking and playing on will never respond well to a very short cut. The same applies to lots of turf as some is often predominently rye.

Also look for shade tolerant species, such as some kinds of turf-type Tall Fescue, or Roughstalk Bluegrass (Poa trivialis)

I don't think the second is grown in the UK but Google shade tolerant Grass and you'll see plenty.

Thanks all, I'll take it all on board.
Just one more question.....Because of the trees I'd like to seed as early as poss. What would the minimum average temperature have to be to give it a chance?

April is the usual recommended time - even it present temps are suitable there is no guarantee that there won't be a sharp frost in March as happened only a couple of years ago. Sowing too early is rarely worth it.