I'd see if friends could help you get the most important areas cleared, maybe a party, and everyone does a little bit - this would allow you to get what would have taken 1 person weeks, based on a couple of hours at a time, and will be a fun thing to be amongst. You can always have an indoors party afterwards.
The cheapest plants would be some divisions or cuttings from friends. Whilst any plant would fill a space,it's ideal if you have some plan ahead of getting everything together. Next cheapest is seed raised, and there are some near-instant colour, from hardy annuals. These are sown now,and grow flower and die this year, but would give you lots of summer colour. Great and easy annuals for lots of colour are Nasturtiums, which come in mixed or separate colours, and are either trailers/climbers, growing 5' or so, or smaller bushes, to 12-18'', and Californian Poppies, 18'', both of which prefer poor soil, which yours may well be, as it's been neglected. Also easy and colourful are Calendulas, 18''-2'. These can all be sown where you want them to flower, and the Calendulas could be spaced out, after germinating, or sown in pots, as the Nasurtiums could too - but not the Californian Poppies.
Some plants to sow for colour next year are the hybrid Foxgloves - Digitalis. These are in mixed colours, and grow 4-5', so are good for backs of borders. These could be pot sown, or sown in the ground, and moved to final positions later in the year - they'll flower early next summer.
These seeds can be bought for less than £1 a packet, often less, such as at Morrisons, or Wilkinsons. They will all re-seed themselves, so you can have lots of flowers for years to come. Just remember not to feed the Nasturtiums or Californian Poppies, as they prefer poor soils. The main thing with young seedlings is to ensure that they don't dry out, as they won't have root networks, to provide enough water, if it's dry for long periods. Once they've grown a bit, less frequent deeper waterings are better, and many of them will be fine without much attention.
One of the problems I would expect is that there will be lots of weed seeds around, and these are likely to germinate and be a problem for some time, so repeated weeding will likely be needed.
As a way to focus attention, away from a larger cultivated area, you could use some cheap tubs, and cram some plants in. They'll need watering and feeding through the summer, as their roots will fill up the soil, very easily. You could use some different plants for these, either annuals, which could be bought as plants, or some that you could raise from seed - the cheapest option.
I'd decide on how much lawn you want to keep, and the cheapest method of improving it, is to over-seed it, with new grass seed - much cheaper than turfing. It'll need watering regularly, to stop it drying out as grows, and then regular deeper waterings that are less frequent, if we get long periods of drought this summer. If you expect your lawn to get a lot of wear and tear, use a blend of grass seed with Rye grass in it - this is tougher, and all seed mixes are blends anyway, but some are more aimed at luxury fine grasses, that can't take as much foot traffic etc. You could over-seed it now, as well as add further lots in the autumn, and next spring.
I live next to Derbyshire, and if I can think of any good suppliers, I'll let you know, but not really any spring to mind atm. Obviously the county's quite large, from Swadlincote, to north of Chesterfield, south to north, as well as into the Peak District etc, and ideally you won't want to travel too far.
My overall advice, is to keep it simple, and not overburden yourselves with too much work, aiming to get maximum benefits from some easy to achieve goals, and then improving things bit by bit. The seed raised plants will be fairly easy, but clearing the ground will be tougher work, that would need doing first, but just a few areas, as well as some tubs - maybe second hand tubs that some friends don't want, would help focus attention and make things look a lot better, than if it's all bare. As time goes on, you'll likely be able to pick up some cuttings and small plants from friends - go for what's easy, as well as fits in. If you had unlimited time, I'd suggest really starting from scratch, and planning it all, but think that could be too big a job for now, and maybe by next spring, you'll have some great ideas about how it could all look.
Here are some photos of the plants I mentioned -
Californian Poppies - Escholzia : http://images.marketplaceadvisor.channel...
Calendula - http://www.thompson-morgan.com/medias/sy...
Nasturtium - http://suttons.s3.amazonaws.com/p/FLSNAS...
Hope this helps. Good luck! Rob