Using QGIS, you can load the shapefile as a new layer, and
Set CRS for Layer to EPSG:7405 (or better EPSG:27700). Then use
Save As... to save it to another filename and EPSG:4326.
Alternatively, GDAL ogr2ogr is the right tool for you.
ogr2ogr -t_srs epsg:4326 -s_srs epsg:27700 dst_datasource_name src_datasource_name
should deliver the data you want, overriding the missing source CRS. You can even batch process a whole folder with ogr2ogr:
for %%N in (source_folder\*.shp) DO ogr2ogr -t_srs epsg:4326 -s_srs epsg:27700 dst_folder\%%N source_folder\%%N
Linux and Mac syntax might be different.