Founded in the 1894 by Samuel Bowley Barnes and Albert Edward Mullins. The partnership started out editing a Banjo magazine called the Jo but soon realised they could use this as a way of selling their own branded product and thus the manufacturing and distribution company was born. 

In the early days I have only seen Banjoleles branded buy them and often called Trumelo and Vaudeville. I don't know if the also branded and distributed instruments from other manufacturers at this point but the must have had their own manufacturing workshop at the time as George Abbott started out managing it.

When production of their own instruments ceased I don't know but in the 60's they were distributing Japanese imported Ukuleles branded Champion, (whether this had anything to do with the US distributor Grossman's Champion brand that was also used on Japanese imports I'm not sure but I think not?), and Banjoleles branded B&M.

The company, (Sam and Al are long dead), are still in business today and still re-branding instruments made by others but using the full Barnes and Mullins name. In the past it was companies like Abbott and Windsor, now it is companies like Dae Won. These days they don't only distribute Banjos and Ukuleles though, they have a full catalogue of instruments they can supply, including wooden Ukulele of all scales from Soprano to Baritone in a number of designs that have traditional sounding names like "the Bowley" "the Calthorpe" or "the Gresse" to indicate the wood choices involved. As well as distributing their own self titled brand of instruments they also are British or European distributors for a number of other brands  like Peavey or Hofner.

Brunswick

This is as branding for the more budget offerings and was founded in 1995. They started out distributing "British Designed"(?) Guitars but have since added Mandolins and Ukuleles to the range. Currently the Ukulele come as Soprano, Concert, Tenor or Baritone scale in a number of laminate veneers plus wooden colourful budget ranges. In the past they have included plastic, (Concert scale made by Lanyao), in their catalogue and some electric guitar homage Ukuleles that look very similar to the K-Wave ones from Japan. They have their own Brunswick website but I'm not sure if they were ever independent

I know they are the main distributor worldwide but I am not sure if they are also the owners of the Lazy brand of budget soprano Ukuleles.