OK, this can be a beautiful space with a bit of work!
First things first... when we get a dry day, give the lawn a mow. If it gets waterlogged, I suggest a hover mower rather than a push one! Don't cut it too close first time around (there will be a lot of dead grass at the bottom - you want to have the mower on the highest setting. Get a garden fork and put a load of holes into it, much as you would fork over pastry for baking. In spring, once the grass starts regrowing, give it some lawn feed and mow weekly working to a progressively shorter cut. In a year, it will be much, much neater.
Get yourself a lawn edging tool (it looks like a half circle on a stick) and neaten up the edges of the lawn, chucking away the extra bits of grass. (Use a piece of string tied to two sticks to make sure you get a straight edge). Combined with a cut, you won't believe how much neater this will make it look.
Give the borders a good weed and perhaps think about widening them a bit - it's hard to tell from the picture, but they look quite narrow. You'll get a better result if they're 2-3 feet deep as you'll be able to plant some wider shrubs or even double plants up. Then get bags of cheap compost and put them over any bare earth. You want to cover it to a good depth - at least 2-3 inches. This will stop weeds regrowing until the spring and will really do the earth underneath loads and loads of good. If it is very waterlogged, you might want to add a bag of horticultural grit to the mix. This will create air pockets in the soil, making it less claggy. If it's really heavy clay, you can also add a bit of gypsum (often sold as a product called 'Claybreaker' in garden centres).
Personally, I'd think about adding another border in front of your low wall at the back of the picture (next to the street), as a bit of greenery there would look really pretty. Just dig out the turf, and put on some compost, as before.
You can think about adding evergreen shrubs now to get you through the winter - things like Mahonia or viburnum tinus will give you some structure and greenery to look at over the winter. When you plant them, make sure you dig a really, really big hole and add loads of compost. But I'd leave most of the planting until the spring now. Have a look at my shade gardening thread - there are lots of recommendations for plants that thrive in dark, wet conditions on there.
Good luck!
www.mumsnet.com/Talk/gardening/2441728-Shady-characters-a-permanent-home-for-shade-garden-suggestions