All the people going on about research etc have clearly never actually done it - talk to people who have, and remember some of the stats about driving in the early hours of the morning will be drunk people, and people who've got stranded so they are pushing on through when they're already tired. Motorway driving at that time is very safe, especially since there is hardly anybody else on the road. Most people I know who live here (EU country) and drive home do it this way. Some stop overnight. But for us that would add an extra day to the travel time as well as the expense - it just makes it impractical. For us leaving at about midnight is the optimal way to do it.
Definitely goes without saying that you must be aware of the tiredness risk and of course stop if you feel tired.
When we drive home from UK we tend to do it in the day instead, so 3h from our families' town to the ferry port, which we get up early and try to do in one go before rush hour.
Then cross, which adds an hour this way, so it's late morning by the time we get over. Try to do a couple of hours then stop for lunch, then another couple of hours then stop for a run around, another hour or so then stop for dinner, and then the last stretch of about 2 hours.
So this is early morning (4/5am) until about 9-10pm. It's OK. It's a stressful day because the DC get fed up. We've done this with various ages of kids (have DC 1.5 - 14yo) and between crawling and about 6yo is when they are the most difficult to travel with.
Doing the journey 100% during their awake time makes it harder because they get bored, so you have to stop off more often and for longer so the journey takes longer. A 10 hour drive takes us about 16h this way. Older children probably do best with this option, though, because they can understand the purpose of the journey. Toddlers just do not understand why they are being strapped into a seat again and they get really angry about it unless they are tired enough to not care or you build in more/longer stops.
Doing the journey during 100% of their usual sleep time (leaving at their bedtime) tends to mean they don't sleep well and then they are really grumpy all the next day. Plus, we found that the adults felt more tired this way, because we hadn't had any time to rest between (hyper, manic with excitement) child wrangling all day and then driving. So arguably you do that "dangerous stretch" of 2-6am when you've been awake for the longest. However it's probably the quickest way to do it. When we did this the whole trip took only 13 hours. There are no rush hours to avoid. We did find they woke up more disorientated in the first part of the journey and then they had long periods awake in the middle of the night, though not as moany as they were when we drove in the day.
Leaving in the middle of the night, so they have had a good 4-5 hours' sleep (and adults get to sleep for 3-4 hours too) means that they tend to sleep through a good portion while the roads are also clear so you can cover a lot of ground. And when you stop you only do minimal stops for a wee and then back in the car and they go to sleep again. This minimises the overall length of the journey. You still have to stop every hour or two in the second half, but that's not as bad. And you attack that "dangerous time" of driving with the energy and enthusiasm of the start of the trip. This usually takes us about 14 hours depending on traffic.
Whatever you do it seems to work to front-load, so do longer stretches in the first half of the journey when everyone has more tolerance and then more frequent stops towards the end of the journey, sometimes with a longer push for the last ~25%.
With positional asphyxiation the "clock resets" when you take them out of the seat, so it's not quite the same as leaving them in a car seat to sleep for the whole night in one go.
You will probably have to change nappies at every stop just because you want to keep DC as comfortable as possible but also because they wee more when they are awake than asleep, and they'll be awake for at least some of the journey.
You/partner will probably end up spending some time in the back with DC.