I would suggest bulk mushroom compost. This is sold locally by the cubic yard for $40 a cubic yard. Delivery costs me about the same price, $40 for 8 to 10 yards, or whatever your sources truck will hold. One yard will cover 324 square feet of lawn assuming you spread it an inch thick which is what I've had experience with.

I wouldn't recommend applying this until just as your lawn starts greening up unless you want a black lawn all winter. You can reseed or over seed before spreading this. You also could optionally aerate the lawn before spreading. It will take about a week for your existing grass to grow thru the MC and it also takes about a week for the quickest germinating grass seed to sprout.

This is called top dressing. It will amend your soil and supply fertilizer for several years. I wouldn't have them drive the truck into your lawn as your trying to improve it. I use a wheelbarrow and .a spade to load it and I either spade it out of the wheelbarrow or dump the wheelbarrow and rake to the desired depth. I use a wider 24 or 30 inch aluminum rake to spread it with, but a standard width iron rake will work.

I wouldn't apply any fertilizer till at least the fall; and I'd go easy on any application in the fall and for at least two years. What would happen is that you won't be able to keep up with the mowing. I'm sure you wouldn't want to mow several times a week.

Any leftover MC can be used as mulch in flower or vegetable gardens.

It should be understood that compost - of any kind - or composted manures are NOT considered fertilizers but only soil amendments. Their nutrient load will always be very low. I'd also wonder how available mushroom compost might be in the UK.

I pointed out that from EXPERIENCE; you'll need to mow twice a week, or more, if you apply other fertilizer to the lawn. As far as your Wondering: Here's a link to the first result on a Google search from the US.

Applying mushroom compost or I assume any compost will improve the lawn for many years. I suggested this one as it's readily available in almost any quantity and I have done this procedure myself. In the first fall after doing this topdressing anyone with a lawn for a few years will be able to judge whether additional fertilizer will be needed.

Mushroom compost is easy to get here. But I’d question why a UK lawn needs any feeding or ’a boost over winter’. Do you want to be mowing all year round? Our grass grows fairly constantly already.

I stated that you should apply the MC in the spring. You want to apply this when the grass will grow vigorously so that it covers up the black mushroom compost as quickly as possible. If your lawn doesn't need mowed and you don't expect it to in the near future; you wouldn't want to apply this then.