Well, on the basis that Vax UK were importing Dirt Devil Swift uprights (and so many others) from the U.S, it gave Vax a bigger market share than they had had previously. TTI ownership thereafter allowed more "Vax" products to be sold.
From previous posts on here, it seems that some Hoover uprights in the U.S under Maytag ownership were worse than previous. TTI aren't to blame for sole destruction of building poorer vacuums - they can built good quality vacuums - but I rather think it is down to economy and cost of parts, one reason alone to why TTI are producing similar vacuums in a lot of the brands they own.
I don't think TTI own Bissell outright but rather share a division of products amongst the brands, in the same way that Shark seem to follow suit - up until very recently for example, Shark's uprights have made it over here under the Morphy Richards brand - and the two companies seem to be developing very similar products. Or just swap and rebadge.