My starting point is that there is no perfect way to parent and it's not really one parent's place to tell the other how to do it - if you're equal parents, you have equal rights to decide your approach.
That said, children thrive when there is consistency - knowing Mum and Dad are singing from the same hymn sheet really helps them understand where boundaries lie, what is/is not acceptable etc. I think the first thing is for you and your partner to learn how to discuss your approaches and attitudes to parenting (without criticism of each other - just open, healthy dialogue). The children are very young at the moment but it won't be long before you're having to make much bigger decisions on what you're ok with. By the time that comes round, you could really do with having a method of communication between you that means you're able to discuss these things and agree a united approach.
As for bedtime itself, does your partner enjoy bedtime or do you think he's deliberately doing a crap job so that you'll step in?
If it's the former, honestly, be glad that the children have a Dad who loves that time with them. Leave him to do things his way (it is his right, after all) and just remove yourself from the situation - take a walk, make a phone call to a friend, sit in the garden, go to the supermarket, whatever.
In conjunction with the above, suggest he just starts the bedtime routine earlier. That's what we do (if one of us wants to extend bedtime - more stories, playing in the bath whatever) we just get going earlier.
Someone much further up the thread suggested the library challenge and I thought your response was pretty poor - you seemed completely closed off to the idea (while you're simultaneously expecting your husband to be open to changing his ways - double standards, much?). Stories at bedtime is one of the easiest way to calm children (and it's obviously great for their development) and I'm surprised that doesn't form part of the routine already. When do you read to them?
You seem to suggest you couldn't do the library challenge with your children because they can't read. How do you think they learn to read if you don't read to them. And aren't you teaching them to read? I'd suggest something structured like the library challenge would be great because if it's "the thing" you're doing, your partner will have an activity to focus (and calm) bedtime with, without it being something you're nagging him to do your way (the reading challenge being an external programme).
As for a getting the phone out at bedtime. Have a look at the impacts on phone lighting affecting sleep. There's a tonne of research on it - ultimately, it's crap for children and adults and really affects sleep.
If you're not able to have those sorts of discussions around the impact of behaviours on your children (and that does not mean pinpointing and criticising each other's actions) then, as I said at the start, you really need to figure that out between you.
My husband and I will do things differently to each other. Sometimes, that's absolutely fine. Other times, we agree we need to be consistent and so we discuss what we think is the best way forward. We don't criticise each other. I certainly don't take the approach that, because I'm Mum, I'm right. He will often do a bit of reading / research (that's his thing) and share it with me, so that can independently help shape our thoughts.
I'd suggest just doing things exactly as you are isn't a great way to go, though, because you're not happy with the current status quo. Having that much angst each night is not a great way to live.
Going back to the questions above, if it's the latter, then that is poor parenting and being a poor partner. That sort of behaviour needs addressing and you need a serious conversation, maybe some couples counselling, about it. Weaponised incompetence is just not something to live with.