11kV/400V TX with no LV protection

You need to realise that protection on the HV side of the transformer is much easier than the LV side, and that transformers are extremely robust when it comes to overload, particularly in larger sizes, or with Oil cooling. Protection of the cables on the outgoing side is also very difficult, where there will usually be several cables in parallel and faults are extremely unlikely. Your transformer HV side presumably has either fuses or circuit breakers of perhaps 60A rating, whereas the secondary may need 1500A fuses. In fact the primary fuses give significantly closer protection to the TX without any difficulty over fault and loop impedance, which may not be as low as you understand from BS7671 and the multiple parallel cables probably will not limit fault currents sufficiently anyway, so the fuse may never fail, just a big burn-up!

You need to realise that protection on the HV side of the transformer is much easier than the LV side, and that transformers are extremely robust when it comes to overload, particularly in larger sizes, or with Oil cooling. Protection of the cables on the outgoing side is also very difficult, where there will usually be several cables in parallel and faults are extremely unlikely. Your transformer HV side presumably has either fuses or circuit breakers of perhaps 60A rating, whereas the secondary may need 1500A fuses. In fact the primary fuses give significantly closer protection to the TX without any difficulty over fault and loop impedance, which may not be as low as you understand from BS7671 and the multiple parallel cables probably will not limit fault currents sufficiently anyway, so the fuse may never fail, just a big burn-up!

I concur with respect to over load but what about short circuit protection. My argument is, if the cable between the outgoing side of the TX and the building it is servign (over 30m away) was damaged, what is providing the protection to that cable? The busbar in the Transformer. Surely the outgoing side of the Tx needs protection?

It does merit consideration. You also need to consider faults at the receiving switchboard upstream of the first protective device.

For a private transformer or similar Interphase fault protection should be achieved by coordinating with the HV protection (fuses/relay). As others have suggested this is normally fine - but someone should check the clearance times and cable/busbar rating.

For a long cable it is possible that the fault impedance will reduce the prospective to the no-trip region of the HV fuse / relay setting, in which case you might indeed need something on the LV spill box

Similar applies for earth faults but if HV protection isn’t likely to operate in a timely manner restricted earth fault schemes will ensure rapid clearance on high impedance faults.

I assume that your transformer isn’t big enough to justify differential protection!

Mitigating the risk by design such that faults are unlikely is a wise strategy regardless.