As far as I know, anyway is a common word used by both American and British English speakers to mean in any case, nevertheless, etc. I never thought much about the word until I noticed that British English speakers sometimes use anyroad the same way.
If I had thought about it, I'd probably have said the way in anyway meant something like manner, or whichever way, or manner, in which the situation is considered.
But learning of anyroad as a synonym for anyway, I thought about the fact that way is another word for street or road. Thus, we can also way that anyway means "any (metaphorical) road you want to go down in considering the situation."
My initial impression was that anyroad was a word invented as an amusing alternate for anyway, unexpectedly playing on the synonymous relationship between way and road, but perhaps there was no humo(u)r intended.
Dictionary.com identifies anyroad as Northern England dialect and British slang, originating 1885 or 1890. What other specifics relate to the origins of anyroad?