I believe there is some misunderstanding between projection / transformation.
The European Grid uses a Lambert Azimuthal Equal Area projection.
The British National Grid uses a Transverse Mercator Projection. UTM is a series of secant Transverse Projections which span the Earth.
Changing cartesian (projected coordinates, e.g. x/y) between projections is no problem as long as the coordinate is within the extent of both projections.
The two projections which you mention use different datums.
The European Grid uses ETRS89 as datum.
The British National Grid uses OSGB36 as datum.
You will need to transform geographic coordinates (e.g latitude and longitude) between the datums. This will always be an approximation.
The picture below shows 100km grid squares reprojected to EPSG4326. Red are the grid squares from LAEA (EPSG3035), blue are the squares from BNG (EPSG27700).
As mentioned in comment a square from one projection cannot be a square in a different projection.
True north (EPSG:4326) will be straight up. The grid convergence can be calculated for any point. But the shape is defined by the projection.
Short answer: Grid convergence is different for different projections but also within a projection.