Thanks for all the responses. I had posted a long explanation straight after the photos but that was lost. What the photos don't show is the patio - access is from the kitchen door and French doors in sitting room.
It's a 4 bed semi on a street of 3 bed houses - mostly terraces. Due to an extension by the original owner in 1970 and a garage conversion, the ground floor is 40% larger than all the neighbours. I bought it last year for £340k. Ceiling price of street is about £300K for a 3 bed terrace though - mine was more expensive as inside had been "done". I'm planning to stay here long term - for the next 25 years until I retire, and then will downsize.
It's just me and my 2 year old DD, but we have lots of play dates in the garden, especially in the summer. My mum and sister have both said my idea to turn it into a gravel garden, with lots of paths is a silly idea because my DD will need grass for the paddling pool and soon will be wanting to put up a tent in the garden etc. I a open to her having a part of the garden to play with - but making the lawn even tinier will make mowing it seem even more a waste of time.
I know that hard landscaping can be very expensive - my budget for the whole garden is about £3 k. From that I would like a new metal shed as the wooden one is falling to bits, and is on bare earth so would need to pay someone to pour concrete as a base, to run electricity out into the garden so that I can have a light over the back gate when I am taking the bins out at night, and to put some things for me into the garden - a lovely bench or arbour to sit and read, a timber disguise shed for the recycling bins on the patio (next to kitchen door) and the wheelie bins, and some lovely plants - I'd like some standard roses, standard magnolia, standard fruit trees and lots of scented climbers, grape vines etc. It's what to do about the lawn really - as at the moment you look out and just see the back fence, and it looks so boxy and suburban and dull...