Are you sure you want lawn?
Its just lawn for someone who says they don't have a lot of time for the garden is rather high maintenance to keep nice and is less practical than other surfaces in the wetter months.
Lawns need mowing every week really Easter - October, they need feeding, edging, scarifying, aerating, being left to recover, being reseeded etc.
In the same time you'd commit to a lawn, you could have some really lovely boarders with a selection of evergreen shrubs and then a few annual bedding plants so you can change your colours and add interest.
Instead of lawn theres gravel, paving, brick/ blockwork, tarmac (don't just think black it can be laid in all sorts of colours), decking or a mixture of any of these.
Regarding fast growing stuff some rambling clematis would cover your nasty fence in their first summer. The problem with anything really fast growing is it wont stop when the fence is covered - it'll keep growing then if not pruned end up top heavy shadding whats underneath, the lower leaves will die back exposing the fence again and the top growth will look like a scraggly mess.
Have you thought about something like a reed screen that you can fix to the fence and then grow attractive slower growing climbers through it?
I know its not what you asked but throws potentially some more ideas into the pot