For old phones the number 3 connection is required for it to ring. However for internet and modern wireless phones it is not required so I would guess wires to number 3 disconnected or wires 2 and 5 swapped wrong way around.
I am sure there are but you can look as easy as me. There are 6 connections in the British plug. The phone line coming in has two wires and the master socket provided by BT (Open Reach) has a number of components that stop damage and store power for the ringing there is also a removable half on socket that allows you to connect up extensions and the first thing is to remove this and plug directly into the socket under the removable bit so it is connected direct and has non of your internal wiring in use. If the phone works in this then not a BT job. Of the 6 connections only three are used. Number 2 and 5 are the real phone connections and many modern bits of equipment only need these two wires. However older phones also use the 3'rd wire to work the bell normally this being wrong will only stop the ring but different makes of phone act different to a fault. The other common fault is the two incomers being swapped over. Again this normally only stops the ringing but depends on make of phone.
Phones that use mains power i.e. cordless often don't use the third wire so even if there is a fault with third wire will still work. If you can borrow some ones cordless phone to try you may find that is cheap way to cure.
At one time electricians would install extensions for phones but with broad band faults were so hard to trace most have stopped doing that work as equipment needed to test when broad band stops working is too expensive for most electricians to want to bother with.
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