If you were asked where this lane is and you weren’t sure of the answer, there’s a good chance you would suggest that it is on the Barbican. Somewhere, that is, on or near the Parade, where you can find both the old Custom House and the more recent Custom House for Plymouth. The former is a delightful old building (sixteenth/seventeenth century?) on the south-side of the Parade, the latter, its successor on the north side, the impressive arch fronted building designed by David Laing and first occupied in 1823.
However neither of these buildings holds the key to the answer. Rather the Custom House in question is the one that once operated at Millbay. A somewhat later and rather distinctive octagonal Victorian building, it was erected just off Great Western Road, which curves around towards Millbay, indeed Custom House Lane runs parallel Great Western Road for a short stretch, alongside the Millbay Marina.
