Not much "lawn" there, lots of moss and annual plants with the inevitable daisies.
First fix the soil, good drainage and aeration. Spiking with a fork to 4 or 6 inches all over. Clay soils and the rubbish left by builders can be a problem. Grass likes a high Nutrient level. Conservation meadows strive to reduce this and thus the competition to the desirable broadleaf species. Scarify spring and autumn, doing this in summer may leave you with bare earth for some time. Overseed with a lawn mix, think about how you maintain and use it when you choose this. I found a tennis pitch mix useful. Don't cut it too short, I maintained lawns at 30mm minimum. That allows easy use without stressing it too much. It allows the roots to penetrate deeper so helping the resilience of the grass. Cut frequently, you should only take off 5 to 8mm at a time, again to reduce plant stress. Feed, not much is needed but it does keep the grass healthy.
Lawns are not easy if you want a show standard, bowling greens ar a full time job. Mine is full of wild flowers and grass, I like the flowers. Because of the drought it hasn't been cut for three weeks, the flowers 30cm high and longish turf, insects love it.
I would consider starting from scratch with a really bad lawn, scrape of the top 3cm, level and reseed. Not a job for the summer. make sure the soil is good.
Rob