Barratts would have wired up the internal extensions to the external NTE. Cannot remember the number of wires each pin can accept, but the older internal NTE has trouble with more than two wires per pin
On the internal sockets do the wires to pins 2 and 5 look like they correspond to the colours at the external.
Openreach will have looked at paperwork and seen that it should all be done, hence no engineer. Getting them to fix things now may cause you to be billed by them, I would make it clear to Barratts that I would want them to take this billing, since at this point it appears their wiring is not working for the telephone.
The broadband may simply be a case of not being activated yet, which provider? Was it a SIM Provider using a LORN?
BTW builders are well known for problems with wiring, there are clear Openreach Builders Guides on all of this.