Hard to tell how many trees you have but please don't just remove trees for the sake of it. That one next to the shed looks really nice, adds character and probably just needs cutting back a little. A lawn without tree's is a pretty boring thing to look at.
Imo, spend the extra and get a petrol mower over an electric. Needn't spend too much. I have one thats about 7/9 year old now, forget the make but its a Briggs and Straton engine and still working well + no chance of cutting the lead. Something like
this mountfield will likley do the job for many years. Don't forget a strimmer (electric or battery I guess) to neaten up the edges and borders.
Pressure washer wise, you could spend £100+ for power but you probably don't need to spend that much as most patio's need cleaning once a year & if you don't tend to wash your own car with one, over spending would be silly. I have one that cost me £40 in Asda 8+ years ago. I only use it to clean up the patio once a year and while it takes a few hours, it does the job well and doesn't take up much space to store. So unless you have other uses for something like a Kerchar might be overpriced for your needs unless you find one on a good deal. <£60-80
Good luck
