Sparse lawn with Woodrush

I would cut the grass then scarify preferably with an electric scarifier (you can hire these by the day). Burn or bin the thatch you remove rather than compost it.
Treat both lawns with an Autumn feed available at garden centres and DIY stores where you should also buy garden lime. Give it a couple of weeks after applying the Autumn feed then spread the lime according to the instructions on the bag. This will change the PH and discourage the Woodrush.
You might want to aerate the lawns as well. It really does depend on how important it is to you to have the perfect lawn.
The Gardener's World website, YouTube, and the RHS website are all mines of excellent advice.
Good luck.

I am not a gardener- what is Woodrush please?

It’s not this is it?
www.theguardian.com/lifeandstyle/2017/may/13/sweet-woodruff-galium-odoratum-starry-flowers

In Germany they use it for Waldmeister flavouring and it is delicious.

No MawB, Sweet woodruff is no relation to Woodrush. The Woodrush which plagues our lawns (well, mine anyway!) is Luzula campestris, also known as Good Friday grass because of its flowering time.

However, it is not a grass, which is how you will be able to distinguish it. While superficially resembling a grass, you will see that its growth form is different. The leaves of Woodrush are arranged in 3 directions (not 2 like grasses) so that the plant as a whole, with multiple whorls of 3 leaves, looks like a many-pointed star. Also if you look along the edges of the leaf particularly towards the base, you will see that they have longish cottonwool-like hairs. Most lawn grasses do not have such hairy leaves.

Thank you - I feared it might be too good to be true to have a sweet herb growing like a weed ! ??

Thank you. I scarified the main lawn, filled the huge recyling container, the resulting grass that is left is yellow and sparse. Today I will try to airate but don't think I've any energy left. If it recovers it will be a miracle. I will apply an Autumn feed but fear it will sit on top as it's so dry and no rain predicted. Who needs a gym when they have a garden. A flat is looking appealing at the moment.

Is it a new build house or built in the past few years?

Peonyrose please follow the instructions on the Autumn feed. It usually states that the product needs to be well watered in if no rain falls within a day or two.
Personally I would water the grass well one day, leave it to dry overnight, apply the feed then after another day or two water it well.
Please don't worry it will grow back over the winter. Always keep your lawnmower on the highest setting and water during dry periods. French it once a week rather than a light sprinkle. Remember that to discourage Woodrush you need to alter the PH. Garden lime will do that, just follow the instructions.
There are companies who come out and do remedial work on your lawn for you if it all gets too much (and you can afford it)

French it should read drench it. I do hate the way autocorrect sometimes changes a word completely!

Well I've had more than enough water on my lawns since ousting, now you can't see the lawn for leaves from neighbouring gardens. I gave to tell myself it's better than the gym.