Reseeding a newly sown lawn. Timescale

I have sown luxury (no ryegrass) seed in the past, and subsequently wished I hadn't. I don't have the time, or inclination, to look after it properly, and it looks manky in drought whereas my rougher grass looks fine.

If you only sowed the grass 2 weeks ago I would immediately overseed with whatever seed mix you think, with benefit of hindsight, you would have used in the first place. It will germinate only slightly behind the seed you have sown, and grow up with it, and you can overseed it in subsequent years to get it to be a higher proportion of the whole.

Having said that I always apply quite a lot less than the recommended amount of seed (as I think it establishes just fine) so adding more wouldn't mean that it got to the point of encouraging damping off / fungal attack etc. If you have been "generous" to start with then sowing yet more might be a worry.