‘I’ll keep you in the loop’ (used a lot at work but is a bit jargon-y)or ‘I will let you know’ - if it is about a specific development or decision rather than just keeping them informed in general! I hope that makes sense! I’m from the UK by the way and I have a job in marketing so talk like this a lot 😂

Maybe "I'll let you know" "I'll let you know what happens."

'I'll keep you abreast of any further news/future info/developments/changes.' But this is very formal and British.

I'll let you know what's happening. I'll let you know if anything changes. I'll tell/call/message/email you if anything happens. These are all possible too.

Not really. These are all sufficient