A normal planning approval could be open to challenge if the LPA mis-directed itself when coming to its decision. For example, if it ignored a valid report, or mis-interpreted its own policy guidelines, if it could be shown that an application would have been refused had the LPA directed itself properly.
However, a Prior Notification is not a planning application, and the LPA has no input if no-one objects. Therefore, an LPA could not be held to have mis-directed itself, if no decision was to be made. It is purely an administrative error and if a neighbour suffers by maladministration (in this case, not being notified by the council) the only recourse is to make a complaint, if necessary to the Ombudsman.
The law is crystal clear that, in the absence of any refusal by the LPA after the 42-day period - whatever the reason - then the work can proceed by default.
As preiuosly, OP's architect should have taken a tougher line with the LPA.