I’m sorry you’ve found / seen this. You’ve obviously been having a hard time and need support and understanding from your DH and MIL.
First things first - are you getting help for your MH. Do you feel it’s working ? If not, get to the doctors and see what else can done to help.
In terms of how to deal with your DH - I’d expect he’s also struggling. A new baby is a lot for anyone, everyone’s sleep deprived and he’s supporting you too. It could be he’s just letting off steam to his mum rather than blowing up at you. It’s better he does this and gets it out of his system than blow up at you. It’s also a lot for at least 3 adults to be living together and I’m guessing the situation has impacted your MIL too. They could just be venting, but I’d be wary of this crossing over into something else with them winding each other up.
However, from what you’ve said it does sound a bit bullying as it appears to have gone beyond just a rant if she’s threatening to sort you out.
Just out of interest - does he feel you are doing what you can to get better ? If not, he might be getting frustrated / resentful as a result.
Can you sit down calmly and talk to him, explain what you saw by accident and how it made you feel ? Explain what you are doing to address your MH and that you want to get better. You’ll need to see what he has to say about the messages.
(Side point you’ll prob get hassle about looking at his phone, but neither my DH or I are precious about ours and from what you’ve said I don’t think it’s really an issue).
Im sure you’re not lazy but MH issues can affect our energy etc... I don’t know if there is any truth to what he’s saying about not washing bottles quickly etc... (although that’s quite a minor thing really - I’m not suffering but still don’t get my washing up done immediately) but I assume they’re done before use ? If not, he’ll have a point that they need doing but if they are he’ll need to relax on his expectations if you are getting it done. Maybe you need to sit together and work out a list of what you feel you can do - it might help you both manage expectations. But it should be done positively and not just be a chance for him to list your perceived failures. You can also tell him what support / help you need.
I know MH issues can be complex, so who knows if what I’m suggesting would work or help. I know some with depression can feel better if they are forced to get up, get moving, endorphins flowing, trying to operate as “normal” as far as possible etc... but that some sufferers find that impossible to cope with.
I’m sure you are doing the best you can for your baby, but part of that is getting your MH in order so you must look after yourself too (which includes the usual fresh air, try going for walks with baby etc).
Good luck.