Between the above, there's not much to add. Planning permission isn't normally required - but might be if they are adding a lightwell or something.
The Party Wall Act is very useful in these circs - that gives your friends a means of controlling what the neighbours do by having to have surveyors appointed by each party agree appropriate safeguards and construction details (at the neighbour's expense).
If they start work without the Party Wall application (assuming they are digging out and not just converting a cellar without changing the floor height), it's unlawful and in theory your friends can get an injunction to stop them (the cost would probably put them off, but on the other hand, bringing it up will stop the contractor in their tracks as they won't want to go there once they realise the work is unlawful).