I've struggled most of my life with bad sleep/insomnia. I read somewhere that some people actually need little sleep but quality is important, thought I'd try consistent early wake time even when I don't feel like it and see if my body clock adapts around it. Basically I get up at 5 am regardless of day of the week. I'm a month in and have been getting up at 5 and going to sleep at 10-10:30, a couple of short wake ups to resettle the little one where needed. No time for naps in my life either, but I don't feel like I actually need them most days. If I'm totally knackered and get the chance I lie down for 10-15 minutes with my eyes closed, alarm set up and that's restorative in a way. Another thing that's helped me is melatonin tablets, round the corner and a few weeks of taking them sets me back on track. I wouldn't otherwise go down the medication tablets, literally noone I know has found them helping long term and they will throw off your natural body clock.
Ran out of characters but I found a book about sleep by Nick Littlehales very insightful. A lot of it a mum won't be able to follow, he was a sleep coach for the Olympics and it's tailored a lot towards sportsmen, but some ideas you can definitely implement.
I had horrible insomnia with my son, also from 6-7 weeks, you can talk to the GP, mine prescribed phenogram as a sleeping aid, it's safe for the first and second trimesters and I used it sporadically just to help with some sleep.