The drain/sewer pipe could possibly be repaired using 'Trenchless Technology', i.e. without digging up floors, knocking down extensions, digging up gardens, etc., etc.
There are a number of companies who specialise in this. The pipe is basically re-lined with a fibreglass liner that is compressed around the inside of the existing pipe. Any protrusions into the existing pipe, such as roots or broken sections of pipe, etc., are removed first with specialised cutting equipment passed along the pipe prior to lining.
The cost of doing this may be relatively inexpensive - and may well be less than the cost of paying the legal eagles to take proceedings!.
If the drain/sewer is in fact private, then the Local Authority really has no responsibility and taking action against them, really is on a path to nowhere. The matter is a civil matter bewteen all the houses the pipe serves.
As suggested above, first get a CCTV survey to see where the problem is (and who may be responsible for causing the damage), then I would suggest getting a quote for the repair of the pipe (by the means of the 'Trenchless Technology' as I outlined above) and pass the CCTV evidence and quote for repair on to the person who damaged the pipe asking them to take action/undertake the repair.
If they decide that it's not their problem and/or they will not take action, well then maybe first a simple solicitors letter should follow explaining the principles of tresspass! By damaging the drain/sewer pipe, they are tresspassing on the property of all the houses the drain/sewer pipe serves as the entire drain/sewer pipe is co-owned by all the properties (same principle as a party wall - it mutually exclusive).