Not knowing you zone or type of grass... you might have to do something different.... but... from my own zone 6 experience with TTTF and a 5x5 area that I leveled in the fall….. there was an inch in the lowest area and less so as it fanned out. I then overseeded as well. Much of the original grass did grown through the soil just fine while the new seed filled in the remaining area. All went well. A few mm of soil shouldn't slow you existing grass at all. How much are you needing to add to areas?

Add topsoil only to fill holes or depressions. Your yard was graded when the house was constructed and you don't want to mess up the grading because you could create drainage problems. If it was not graded properly, then you should get professional help to fix it. It's important for water to drain away from the house.

Missing some key points. If you rototill first you will be left with an uneven subsurface where the tiller hops up out of the ground and where it digs in deep. It hops up when it hits a root, rock, or dry soil. It digs in when it hits a moist spot or sandy spot. When you level the surface over top of the uneven surface, you have differing depths of fluffy soil. Over the next three years the fluffy soil will settle down and match the underlying unevenness resulting in a bumpy surface for the grass. Ideally you will begin with the current surface and simply fill the holes and scrape off the high spots. So that's a very important point you were missing.

Laying soil over existing grass is not problem free. The taller the grass the harder it is to get the new soil smoothed out. Grass gets in the way. You are correct that it is easy to get too much soil over top of the grass and kill it. Mowing it very short sometimes helps this, because the stemmy grass is more rigid and does not get bent over by the new soil. But some grasses don't like being mowed very short, especially all at one time. You could gradually lower the mowing height until the grass adapts to the lower height and looks sort of normal down low. Then let it go 2 weeks and mow back at the low setting and apply the sand or soil. Topsoil has issues with clods, but if you already have it, then deal with the clods.

Another approach, if you know where your low spots are, is to mark those and fill them a little bit at a time. Maybe 1/2 cm per week at the most. Usually these low spots need to be feathered in. They typically start as a larger low spot, maybe 1 meter across, and then in the center they fall off as much as 5 or maybe 10 cm. You can fill the 5 cm hole quickly, but you might want to bring up the entire 1-meter across low spot. The best way to do that is to remove all the grass and start over, but I don't think that's what you want to do. There will be some compromises on your part.

Are you the person from London?

I'm from Oxford, UK. Thank you all for the advice. I got impatient and followed this...

http://www.rolawn.co.uk/how-to-seed-lawn

I need the exercise and spent three hours digging over the lawn which is about 26m2. I burned 1800 calories according to my tracker! Having read the above I can appreciate that levelling over time is now going to be an issue. I dug really deep in some places :-( I did however recently buy the heaviest roller I could afford, so hopefully this will help to minimise the problem.

It's dark now, but tomorrow I'll post a photo of the bomb site I've created and you'll laugh!

However, I've got three more areas to do of a similar size and it sounds like I'll be better filling in the gaps with topsoil and smoothing out the raised areas. This will save a lot of time and I'll head back to the gym which will please my training partner.

Thanks again.