This is a picture of the where the lead soil pipe connects to the CI stack which I'm very reluctant to mess with for fear of creating/finding a bigger
problem than we have already.
That`s an unusual connection , with the stub of iron pipe out the tee , but it works in your favour for a DIY repair to start with . . First measure the diameter of the lead pipe next to the stub that`s pipenailed to the wall , then get a Fernco coupling for that size to 110mm plastic soil . Next remove WC and start cutting out brickwork around pipe on outside wall - when you can see the dodgy connection in the wall , go inside and pull out the plastic soil - leaving a neat hole in the tiled wall . Cut the lead near to the nailed "ears" of the iron stub - use a cheap carpenters throwaway handsaw . Connect it all up with plastic through the wall and onto the WC outlet . Worst case scenario is the fernco leaks a tad onto the flat roof , it`ll seal up eventually
The soil stack looks like a light cast one - so best left alone or replaced complete - but this could be done later from the repair you do now