I started off listening to talk radio stations in the UK on my clock radio with a sleep timer when I was a teenager in the '80s, and I see listening to podcasts as an extension of that.

I use Podcast Republic on my phone linked to a set of Bluetooth sleep headphones similar to [1]

In terms of what I listen to, it's mostly factual podcasts from BBC Radio 4, or providers such as the Naked Scientist or History Today.

[1] https://www.amazon.co.uk/Sleep-Headphones-Bluetooth-Ultra-Th...