Another approach is to consider: Building Control sign off completion, (but design and planning conditions are not their remit) then council tax is triggered. After this, as I understand this you can make planning applications. So it won’t stop your progress as such.
Enforcement from the planning dept are the people who could turn up to look, but usually only if a neighbour complains. But you are waiting for materials, so what can they do?
It’s a bit of a game of chess.
Can you find your own heritage advisor? Or look at other local planning applications for ideas?