Posted: 18/08/2017 at 15:28
For now - I'd just cut that - don't take too much off the first time as it's quite long. A week later cut again a little lower to get it to the best sort of height. Depending on where you are and what weather you're getting, you could either dig out the dandelions (which seem to be the main weed you have there) or use a bit of spot weedkiller on them. Ideally you need a bit of growth on the weeds for it to be effective, but be careful of getting any on surrounding planting or the grass. Wait for a calm day, and preferably till later in the evening when there are fewer pollinating insects around.
Since it's newly seeded, it's probably best to leave anything else till spring. You can then start a programme of scarifying, aerating, feeding etc. Just keep cutting regularly, only taking a little off at a time to keep it healthy, until growth slows up for the winter. I'd agree with the others that it'll be difficult to keep grass thriving under the tree. If you don't want anything too high maintenence, you could simply clear a circular area round the tree and dress it with bark. An edging strip of some kind will help prevent the bark 'travelling' onto the grass, and make mowing easier too. Some spring bulbs would be attractive in there, and that's something you can do in the next month or so. I'm not keen on serious underplanting of a mature tree like that, as there's even more competition for water and nutrients, and it can just look messy, but that's a personal opinion.
If you can/are allowed , remove all the excess stuff on the main trunk of the tree - that'll make a big difference to it. You might be allowed to raise the canopy as well. That just means removing some lower branches, which would allow a lot of extra light into the garden. That will benefit your grass. I expect a shade mix of seed has been sown, and that's the best type for your conditions. You'll just have to play it by ear a little.
Hopefully you can maintain a nice green space with the tree as an attractive focal point. It will take a bit of work, and you'll probably need to keep the grass slightly longer than the ideal because it will grow less well in the shade you have. Good luck with it anyway.