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I think the first step in understanding this is to use an authoritative tool like Exiftool or ExiftoolGUI to extract the actual GPS location recorded in the exported file's metadata -- this will indicate whether the issue is with LR or the other program.
ExifToolGUI Metadata GPS - North 51.115467° West 4.227883°
So you entered 51°6'56" N 4°13'40" W in LR, and Exiftool shows that's what got stored in the photo's metadata:
51.115467 N= 51° 6' 56" N
4.227883 W = 4° 13' 40" N
Thus, it appears that the problem is with the other program -- what is it?
The program automatically entered the GPS data once I dropped the photo on the map. As for what I've tested it with so far:
Irfanview -> Information -> EXIF Info -> Google Earth (Correct Position)
Irfanview -> Information -> EXIF Info -> Show in GeoHack-wiki -> Google Maps (Wrong Position)
Synology Photo Station 5 -> Tag -> Geotag (Wrong Position)
51° 6′ 0″ N, 4° 13′ 0″ W
Photo Station Information
Personally it seems various programs etc are slightly adjusting the geotag slightly.
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One hypothesis: the Google Maps API seems to have two different ways of showing a coordinate location: the precise location and the nearest named entity (street, town, etc.). Perhaps some of these programs are showing the location of the nearest named entity, rather than the precise coordinates. See this thread for an example:
Sorry I typed Google Earth info wrong, should of been Lat:51.115501, Lon:-4.227833.
I guess it could be but having said that when going via GeoHack->Google Maps it places the Green arrow in the wrong place with another point highlighted by Red A Balloon on the closest road. Going via Photo Station just results in a Red Balloon being dumped where GeoHacks puts the Green arrow.
To sound out your idea about nearest named entity here is another example, I took 6 photos from the same position in Richmond park yesterday and LR4 shows this as 51°26'6" N 0°16'59" W and if you enter this in Googlemaps you'll get the green arrow appearing in the right position.
Exiftool Info - Lat 51.435075°, Lon: 0.283162°
Inrfanview -> GeoHack -> Google Maps = Wrong Position http://maps.google.com/maps?ll=51.433333,-0.266667&spn=0.01,0.01&t=m&q=51.433333,-0.266667
Inrfanview -> Google Earth = Correct position (Lat:51.435001,Lon:-0.283167)
Photo Station 5 -> Geotag -> Google Maps = Wrong Position (51.43, -0.27) - Matches Geohack's position.
In the case of the wrong position if we work off the nearest entity then there are about 7/8 closer than Broomfield Hill Wood where the position ends up. I think you are right in the API idea since I would suspect at the very least the one in Synology's NAS Photo Station Package might be a little bit behind the latest.