One of the things I learned on an EMC and lightning protection course is that when driving some of the tunnels through the Alps, black powder fuses rather than electric detonators had to be used, as the fairly frequent lightning strikes on the mountain above would be sufficient to set off an electric detonator.
The problem is that the rock is not particularly conductive electrically, and when a pulse of up to 400,000 amps is applied, large voltage gradients arise throughout the mass. It is possible that the lowest-resistance path to the general mass of the earth may be through some of the equipment in the tunnel. If not, there is radiated coupling to think about, as the rock provides poor screening.
No risk at Boulby of course. Sea water attenuates at, if I remember correctly, 180dB per metre, so the radiated component will quickly disappear. Pure resistive effects will cause a possibly lethal voltage gradient in the water for some distance, but at the sea bed, and below lots of rock cover, I doubt that there is an instrument sensitive enough to detect a lightning strike at the surface, except by whatever is induced in the inevitable power cables etc, which run from the surface.
But in a thunderstorm, you can't necessarily enter an old mine or cave and obtain complete protection from lightning. If there is lots of conductive cover (soil and clay) it will be ok, but if in rock with low water content, probably not. But it is still somewhat safer than being out in the open.