Age UK City of London is a charity working in the City of London which can deliver emergency supplies such as food, toiletries and cleaning products. They arrange activities and can check in on you. That looks like a very good place to start.
Then there is City Connections. It has the most confusing maze-like website. But if you start on this page, you will see that they offer advice on every kind of problem. This is provided by Toynbee Hall, the well respected local charity for people who need help. You can go personally to advice surgeries held at Golden Lane Estate or the Neaman Practice just off Long Lane. (During the Pandemic personal visits are suspended, but you can talk to Toynbee Hall on 020 7392 2919 or email firstname.lastname@example.org)
The Neaman Practice is the Barbican’s ‘local’ GP surgery. You might consider signing up with them if you are moving into the Barbican Estate.