Location of Openreach modem and Home Hub (BT Infinity 2)

In my previous property I asked the BT engineer to locate the modem/HH in an upstairs study and this was no problem, no extra cabling required (master socket was in kitchen) all extension sockets still work fine.
I moved home and had BT Infinity 2 installed, phone line enabled same day. The house was built in 2007 so has extensions in all the rooms. Looks like the master socket is in the cupboard under the stairs. Although the previous occupier obviously had the openreach modem and home hub located in the lounge, as the mounting screws etc were still there. Here none of the extensions work, the engineer said this is a side effect of moving the master socket. But why was this possible in my previous house?

Just a guess: In the old house they installed a special FTTC extension from the master to the upstairs study (high quality cable, filtered for FTTC). The other extensions from the master remained unchanged (now filtered for phone). In the new house they installed the master socket elsewhere and disconnected the old master and this the extensions connected from it.

When I had Infinity installed a year ago last March my master socket was moved by the BT engineer from downstairs to upstairs. All the 3 extension sockets I have in the house continued to work after the move.

Sounds like the engineer wanted to get away. There's no reason why the extensions shouldn't work.

Spoken to BT who said they can send a technician to fix this, but will be chargeable at £130. Annoying as they should have sorted during the rushed install.

Not sure from your original post. Were the extensions working before the engineer installed FTTC on his visit? If not then BT are not responsible for anything beyond the master socket and the FTTC install does not include making existing extensions work. Sounds like the extensions were possibly disconnected before you moved into the house and so are your responsibility not BTs?