[Thank you both for taking the time to reply in so much detail for me
I've ordered a big bucket of general lawn fertilizer so I'll start by feeding the entire lawn and give it a week or so for the weeds to grow back up a little. I've also now ordered some Verdone Extra so once the fertilizer has done its job I'll give the entire lawn a good dose of Verdone and see how it goes and if necessary I'll give it a second dose after a few weeks have passed.
I have a rotary mower so I'll put that on the highest setting for the initial cuts and see if I can get the existing lawn to recover some before I start the landscaping work.
I've got a plan in my head for the garden layout but drawing it out sounds like a good idea. I thought I'd do a large shrub bed at the bottom of the garden along the full width of the fence as when you look out of the kitchen window this is the part of the garden you see the most and it also has by far the best quality of soil as the garden slopes downwards slightly so the soil here stays moist.
I was intending to put in two large curved beds, one in each corner, with gently inverted curve joining the two together across the width of the back.
I'd quite like to plant something in the back left hand corner to act as a screen too as behind the garden is a very large block of flats and their windows overlook the house and garden but I'm unsure what I can plant in a small garden without it leeching the goodness out of the soil and overshadowing the rest of the garden too much. It needs to grow to around 10ft tall ideally to offer any kind of decent screen.
I thought I'd also run some narrower straight beds up the sides of the lawn along the fence as I hate having grass growing up the side of the fences - it's a nuisance to cut and it's a bit boring really.
The left hand side will be taken care of with the slabs and a new 4x3 shed that will sit on there. There's a brick shed near the house so I don't need a second huge shed, just a little one will do
Behind the photos I posted there's also a patio across the full width of the house, although it's not very large but I hope to expand it further down the line. For now though it takes care of any patio requirements
I'll pick up a gardening fork as I don't have one anymore (got lost during the move) and see if I can get some drainage into the lawn as you suggested. The soil around here is quite heavy clay so drainage issues are pretty common but the slope of my lawn does help it to run off better than if the lawn were completely flat.
I guess once I've got the existing grass a little healthier I can transplant the turf from the bottom into the grooves where I removed the path as you suggest and then I'll cut in the shrub beds and level out any remaining severe dips in the lawn with some topsoil and overseed.
I just hope I can manage to get it all done before the growing season is over but if not then at least I can get the landscaping work done and get the plants in as the weather starts to cool. If necessary any seeding work can wait until the spring, as long as the bulk of the work is done

You've given me some food for thought and some hope that I can get it all in decent shape without having to do anything too drastic.
I'm not too worried about the lawn being 100% moss and clover free, I'd just prefer there to be a little more lawn than there currently is but if it turns out that the moss keeps coming back due to drainage issues, even after I go to town on it with a fork, then I guess it's easier not to fight an uphill battle with it as without digging the lawn up and putting in proper drains it's unlikely to be a fight I can win 100% As long as it's all green and the broad leaf weeds are gone, I don't mind that much - I'm just after a family friendly garden in this case so it doesn't need to look like a bowling green.

My mum works in a garden center so she gets a lot of plants for free when they're throwing out stock that's been on the shelves for too long so I've amassed quite a decent collection of shrubs that are all sat in pots waiting to go in so I'm pretty lucky in that regard, I'm not going to need to buy too many plants at all to kit the garden out once the beds are dug

Thank you both again; I'll try to make sure I keep this thread updated as things progress!