Welcome Tango Charlie!
I'm from your side of the pond, and further south (Athens, Georgia). Allotment-Garden is a great place to learn a lot about all things related to gardening.
Regarding your lawn questions, as someone that has recently needed to re-establish a lawn, I may be able to help, but I have some questions for you.
- Do you know what type of grass you have, or what you would like to establish (fescue, centipede, bluegrass, etc.)? Or would you call it a mixed lawn (just looking for green, no bare spots/manageable weeds)?
- Have you run a soil test, or do you at least know if your soil is acid pH <7 or basic pH>7? The pH is important because that governs nutrient utilization. If you don't have that information, you can probably get a soil test done for a reasonable cost, try contacting your local county agent or cooperative extension service office (ask at the closest university with an agriculture department if you don't know where to start, or look online). The best soil lab I have ever worked with is Agro Services International, in Orange City, Florida, their basic test is quite economical, and provides a lot of information. I have also seen mail-in soil test kits at big-box home improvement stores (don't know if those are useful or not). I can provide some instruction with how to collect a proper soil sample if you need it. I am guessing from where you are located that the soils in your area are developed on sandstone, limestone and shale. (I'm a geologist BTW). If you are on the sandstone, your pH is probably acid, limestone, probably basic, shale could be either acid or basic.
- Very generally, what is your soil like? Dig a shovel full down to 6 inches or so. If you can, take a photo of that fresh broken soil to post here, the color provides some information on whether the soil drains well or sometimes stays too wet. Grab a small handful of that soil, add a little water if its dry. (A) Squeeze it, does it completely fall apart (sand), or form into a ball? (B) If it can form a ball, now try flattening it into a ribbon (wet it with more water, if needed). (C) If it can't form a ribbon at all, it is a loamy sand. (D) If it forms a weak ribbon no more than an inch long, it is a sandy loam or silty loam (sandy loam feels gritty, silty loam feels smooth). (E) If it forms a ribbon up to 2 inches long, it is sandy clay loam or silty clay loam. If it forms a ribbon more than 2 inches long, it's a sandy clay or silty clay. (F) Finally, if it feels neither predominantly gritty or smooth, but forms a ribbon either 1 inch, or up to 2 inches, or more than 2 inches, it's a loam, or a clay loam, or clay (respectively). A full guide on how to do this test is at nrcs.usda.gov/wps/portal/nrcs/detail/soils/edu/?cid=nrcs142p2_054311
- From your description, it sounds like you have both a disease problem, with the moss indicating either too much shade, or too much moisture, or not enough nutrients. The mushrooms may not be a problem. With respect to the moss, do you have an impression whether too much shade, moisture, or nutrients could be the problem?
- For the disease issue, I strongly recommend a product called Scott's Disease Ex, it is widely available, and though not the least expensive product, it is economical because it works better than many products that are less expensive. Use a spreader to apply any disease treatment evenly, according to directions. Disease management for prevention works best when the grass is actively growing. Disease is usually worst under hot and dry conditions. If you water your grass during the growing season, water in the morning to allow drying before nightfall.
A soil test will help you identify any nutrient deficiencies. It would be better to provide information on how to fix those problems with a little more information in hand. Go light on fertilizer (lowest application rate) until a test tells you what you will need. I would not add lime or gypsum until you know something about the pH and any deficiencies. All grasses use a good bit of fertilizer, best applied in smaller doses spread throughout the growing season.
This is a good time to plant some grasses (like fescue), but a bad time for other grasses (like centipede). Some close-up pictures of the grass you have could be helpful, even if the grass is dormant. Advice for how to re-plant depends on how much area is dead/damaged, and the type of grass.
Probably too much information all at once . . . but I hope it helps.