As a rule of thumb, most metamorphic and igneous rocks  as well as densely packed mud- and sandstones are poorly permeable - for as long they aren't fractured.

I didn't know before, but I see that the Chunnel was drilled largely through chalks from the Cretaceous, and the heavily fractured chalks on the French side caused a challenge to contain the influx of high pressurized water. That certainly does not sound ideal, though I guess the technology then was already available to sort that out.

As for the geology of the bedrock in the Irish sea, it indeed looks like it's older - mostly Triassic, Permian and Carboniferous sand- and mudstones, with some younger igneous intrusions. I'll repeat that I'm not an engineer, but from a first perspective, that sounds nearly ideal compared to drilling through chalks. Chances are high that the rocks will only be lowly permeable.

Furthermore, if you follow the link and look at Figure 2, it looks like the lithologies are continuous underneath the sea. That also sounds a lot more ideal to me than tunneling through a heavily fractured and compacted mess, like the Gotthard Base, which sounds like it was practically packed with geologic challenges.

So at the back of an envelope, chances and cost risks actually look pretty decent for a tunnel underneath the Irish sea.