11kV/400V TX with no LV protection

I agree with those above regarding how it should work, ie that protection provided via the upstream HV protection should suffice (and indeed is necessary to protect the LV windings), and a short section of LV tail would be expected to be covered by operation at HV. However I would note that the OP has suggested the LV tails are ~30m long, which is a little longer than usual (but by no means problematic) and going between buildings suggests not in a "no fault zone", and furthermore the implication is that selection of the cable is within their scope to select. One ought therefore to consider whether the HV fuse/relay provides sufficient cover at LV through to the first piece of LV protection, and whether the cable will withstand the fault.

It is also worth noting that some HV protection does not operate on low overcurrents; for example HV transformer fuses are sometimes specified as backup rather than full range protection (to avoid inadvertent operation on LV faults) while self-powered relays such as the common Schneider VIP range require a certain current through the CTs to power up the relay. Hopefully this will have been considered by the HV designer in the selection but one shouldn't assume and if you have a long run to the first piece of LV protection you ought to check you have enough current to operate the HV at the far end of the cable. In most cases it will be fine.

I agree with those above regarding how it should work, ie that protection provided via the upstream HV protection should suffice (and indeed is necessary to protect the LV windings), and a short section of LV tail would be expected to be covered by operation at HV. However I would note that the OP has suggested the LV tails are ~30m long, which is a little longer than usual (but by no means problematic) and going between buildings suggests not in a "no fault zone", and furthermore the implication is that selection of the cable is within their scope to select. One ought therefore to consider whether the HV fuse/relay provides sufficient cover at LV through to the first piece of LV protection, and whether the cable will withstand the fault.

It is also worth noting that some HV protection does not operate on low overcurrents; for example HV transformer fuses are sometimes specified as backup rather than full range protection (to avoid inadvertent operation on LV faults) while self-powered relays such as the common Schneider VIP range require a certain current through the CTs to power up the relay. Hopefully this will have been considered by the HV designer in the selection but one shouldn't assume and if you have a long run to the first piece of LV protection you ought to check you have enough current to operate the HV at the far end of the cable. In most cases it will be fine.