Turns out that there was no real "Edison Lighthouse." That was the name that the writers and the session singer Tony Burrows (yeah, I never heard of him, either) used to record. However, when the song became an overnight sensation there was no band to go along with the song. Tony Burrows appeared on TOP OF THE POPS to do the vocals but a band called The Greenfields were hired to pretend to play behind him.
After the song hit its peak of popularity, the group quietly faded away.
Until 2001, when the movie SHALLOW HAL used the song on its soundtrack. The extremely obese character played by Gwyneth Paltrow was named "Rosemary."
I was going to use a clip from SHALLOW HAL but because the song was used as the closing song, it ran over the credits. That's boring, so I pulled up this obscure video of Edison Lighthouse's "live" performance from TOP OF THE POPS from February, 1970. The vocal is probably by Tony Burrows, who is at least singing. If you watch the guitarists, however, they don't appear to actually be playing.
I have no idea who the initial men are or what they are doing for the first minute. Skip that if you just want to see the song.