It might be easier to georeference the tif to the points.
Not familiar with the British National Grid, but it sounds like although you may have projected the points, their values are still in their original system (GCS?). So if your coordinates are 51.5 N 0.4 W, your point will be 51.5 meters N, 0.4 meters W of whatever the origin point is for the BNG (assuming its unit is meters). If all your points seem bunched up into a small area, this is most likely the cause. Did you define a projection as well as project the points? One quick fix that may work is to create a feature dataset in a file gdb, set its coordinate system to BNG, and import your points into that.