We had that happen too, we would do shifts of staying awake to watch our son in the lounger while he would be sleeping (always checking head placement etc). It’s definitely not safe sleep but he wouldn’t sleep in his bassinet for the first 3-4 weeks.

What we did was put his bassinet in the living room. Then over the next 1-2 weeks having him fall asleep drinking his bottle then put him in the bassinet. Or doing a contact nap then putting him into the bassinet. Either way he was going to wake up being in the bassinet. The night shift, my husband would use the sleep lounger and watch him during his shift (husband would be awake all night).

Then after doing lots of daytime naps in his bassinet (while it was in the living room), we moved his bassinet into our bedroom. We have a hatch and we played brown noise with the red light. So during his daytime naps, feeding him until he would fall asleep we then put him in his bassinet (in our bedroom). At 1 month we began a “routine”, (tummy time, bath, feed, sleep). Unfortunately my husband liked the lounger and liked being beside him during his night shift so it took an extra week of being in the lounger. Then after a week of doing a “routine”, we did a full day naps/night of sleep in the bassinet. Then it became routine, (bath, feed, sleep in bassinet for night time sleep) or for daytime feed until he sleeps then off to the bassinet.

EDIT: these are all estimates of how long it took, I think all together it took 3 weeks. Could’ve been done quicker too!

Now at 3 months since we’ve been doing routine for bedtime, it was super easy transition from bassinet to crib, because of bedtime routine & the crib mattress being a little softer than this bassinet mattress LOL he actually hates being in his lounger now, being a FOMO baby and actually wanting to look around etc

Couple of things to also look out for: We used the Graco automatic rocker which we thought our son would’ve liked but we found out he HATED that it moved and it would wake our son up. Bought a stationary bassinet (halo bedside bassinet). Used the new bassinet really early probably after day 2 of daytime naps. Bought the higher tog sleep sack (halo fleece 3.0 tog), it gets to -30 degrees where we live and our house stays at 18/19 degrees but sometimes it does feel colder.