Hi Sue,
I only have personal experience from my overgrown lawns, but have been astounded at how quickly they have turned from that to nice meadows. Essentially, I have just been managing them as such, with one small application of seed hand harvested from a very nice local meadow. This year, one has to be at least 50% Yellow Rattle, with lots of Common Vetch and some knapweed. The other is more variable, with still some rank grasses in places, but increasing amounts of Yellow Rattle and other nice wildflowers.
I am lucky that we are next to grassland with anthills, so the ants have already started moving in too. This has been quite interesting as I was always told that only very old meadows had anthills in them, but the size of some of the ones in my lawn is already quite amazing and this is the sixth season of managing it this way.
As well as my two ‘meadows’, I have an area of grass where we cut some of it, but mow round the nice flowers through the growing season, leaving stunning patches of buttercups, clover, stitchwort, etc. All adds to the diversity. 🙂
Tammy