I’d be looking to address the root cause of why all the grass died before re-seeding. If you just re-seed again you’ll likely get the same result.
Based on the picture, I’m guessing it’s a relatively new build and the garden was waterlogged/boggy in the winter?
I’d dig a small hole to see how much top soil you actually have before hitting what I would suspect is compacted clay underneath which stops the water from draining away. To get a reasonable lawn going you need to sort that out.
The lawn was all overgrown knee high when I moved in, after they cut it once it all died, also I didn't realize I had to water it in summer.
The gardener already told me yesterday the soil is terrible and ideally needs new compost and turf which would cost around £500. The rake and seed was £26. So the idea is to try that, dig up all the soil and get it loose and aerated first, and if necessary buy some extra mulch myself to save the difference from £500.
The neighbours lawn is growing fine and is all connected under the fence, so it should be doable if I simply add doing my own maintenance.
It was built in early 2018 and I bought in September on Shared ownership, and only have £27k left to pay on my share, however inflation means the remaining share price will go up so prolly just leave it being paid for with housing benefit.
Going to order a spade and fork to make the rest a lot easier, but I'll still need the gardener to mow and trim the hedges once or twice a year.
This was the best priced metal set I found on Prime delivery
https://www.amazon.co.uk/dp/B0817K9XKT?psc=1&smid=AROLF024UQVE5&ref_=chk_typ_imgToDp
I nearly bought a plastic set for a fiver.
Also I have 2 trees that are still alive along with all the hedge, theres the one in the garden, and a medium height one at the corner of the border. The hedge has those yellow plants that are some kind of 'bloom', and a couple of lavenders on each side of the path to the gate.
I just tried raking it a bit more and like a dummy realized I had stopped earlier cos I needed hand tools to remove the weeds easier. Should be getting them tomorrow.