So let’s pick up on the first of these in detail…
Net worth and manually connected accounts
This strikes me as fairly straightforward and can be done by a couple of tweaks to the accounts screen (the one you get from pulling down on the card on the feed).
Firstly, I’d add a net worth balance at the top of the screen, adding up the connected accounts.
Secondly, I’d let users add manual accounts. So for the ones that can’t be connected by open banking you’d be able to enter the current balance manually. Excellent for tracking things like mortgages etc. Bonus points for being able to store history.
Thirdly (and this is more complicated) I’d add a bit of intelligence. Let me enter the amount I have on a loan, find the payment from Monzo that repays it, enter the interest rate, then have Monzo approximate the current balance based on transfers from my current account. This would also work in the inverse way for savings accounts. Pick your savings sort code and account number, set the current balance, and have Monzo automatically change your savings balance on the basis of transactions in and out of your Monzo account. No connection needed.
What might this look like? I’m glad you asked!
Only one other thing to add: I’ve taken the liberty of having this as a separate tab. Anecdotally, the discoverability of this screen is pretty poor. And by making it more important (effectively the home page to your financial life) I thought it definitely deserved its own tab.
More 
. In the meantime, thoughts? Is this going on the right lines?