It's definitely spelled Londonlike. KBB introduced the model in 1917; an issue of the American Exporter at the time stated "The shapes in this line are identical with the popular London Styles, and from this fact the name
Londonlike is derived." Makes sense given the cachet associated with London made pipes. And a refreshingly honest approach. Some other American manufacturers were less candid in differentiating between what was inspired by London brands, and what was actually of London manufacture.
Here's a 1919 advertisement mentioning the model:
It's clear that the Londonlike was available in a smooth finish too.
Here's a poker:
And a cased beauty:
I think, but am not sure, that it was produced with vulcanite as well as bakelite stems. All mentions I found of the Londonlike were from the late teens. If it survived past that point I would guess not very far into the 1920s. Bakelite as a stem material lost favor as the decade waned.
In any case yours is a very cool pipe and I hope it gives you much pleasure.