Man of Honour
Do the military automatically deserve respect? Some might disagree with that. But that is a different conversation. You are perfectly within your rights to video a military base from a public place. That includes American military bases in the UK. You can expect to be challenged for doing so but they have no powers to arrest you unless they can prove malicious intent.
What complete and utter tool would want to do that?
For what reason?
Same as filming the Police walking in and out of a Station, why?
Do you really think it's a worthwhile project?