Arthur Mountbatten is not tied to any main story mission unlike Kiera and Mad Jack. To recruit Arthur Mountbatten as a companion in Fallout London, you need to travel to the northeast corner of the map near Hackney.
How To Recruit Arthur Mountbatten In Fallout London
Head north from the Tower of London end of Tower Bridge and make your way to the large concrete wall block at the very top of the map. It is not very hard to spot but if you have trouble refer to the image I shared above.
You will find Arthur Mountbatten hanging upside down from a tree just in front of the Hackney Checkpoint. He is not easily noticeable due to his position so you will have to go a bit closer to be able to see him.
Speak to Arthur and he will ask for your help. Cut him down from the tree.
After helping him, Arthur Mountbatten will agree to join you as a companion. He uses a .45 Revolver and I have to say he is very handy in combat.
