As you discovered, @TimG, Google was giving you the Irish Grid coordinates. BritishNationalGrid.jl only does OSGB NG coordinates (and does them correctly), which explains the apparent discrepancy.
The Irish Grid is described by code EPSG:29901, so if you wanted to convert from BNG to ING with Proj.jl (which BritishNationalGrid.jl uses anyway), you could do this:
julia> import Proj
julia> ing_from_bng = Proj.Transformation("EPSG:27700", "EPSG:29901", always_xy=true)
Transformation unknown
source: OSGB36 / British National Grid
target: OSNI 1952 / Irish National Grid
direction: forward
julia> ing_from_bng(148376.87358622215, 531260.3815596306)
(335811.9954414863, 375965.6198268878)
To do the transformation the other way, just reverse the EPSG codes.
Note, however, that this will not be accurate to more than a few m because the high-accuracy grid of corrections used in transforming to/from BNG is only valid over part of the island of Ireland.