Read https://www.gdal.org/gdalwarp.html carefully
-te xmin ymin xmax ymax:
set georeferenced extents of output file to be created (in target SRS by default, or in the SRS specified with -te_srs)
What is wrong that you have given xmin=51.50386 and xmax=-0.4333941. This should work better
-te 51.50386 -0.4333941 51.50391 -0.4333865
However, if you are close to London you have the coordinates in wrong order. GDAL wants them always as longitude-latitude or easting-northing order so the parameter would come
-te -0.4333941 51.50386 -0.4333865 51.50391
But then you should re-read the manual https://www.gdal.org/gdalwarp.html "(in target SRS by default, or in the SRS specified with -te_srs)". Your target is EPSG:27700 so either you must give the extents according to that or if you want to select the area by lon-lat coordinates you must tell that by adding the -te_srs parameter
(GDAL >= 2.0) Specifies the SRS in which to interpret the coordinates given with -te. The srs_def may be any of the usual
GDAL/OGR forms, complete WKT, PROJ.4, EPSG:n or a file containing the
WKT. This must not be confused with -t_srs which is the target SRS of
the output dataset. -te_srs is a convenience e.g. when knowing the
output coordinates in a geodetic long/lat SRS, but still wanting a
result in a projected coordinate system.
-te_srs epsg:4326 and it should finally work.