It won't, unsurprisingly the plastic straws that get disposed of in the UK will either get recycled or get crushed and stuck on a landfill site. The plastics that end up in the ocean generating the viral images of sea life clutching straws etc.. tend to mostly come from various asian countries.
Various celebs who regularly travel by private jet, have parties on yachts and travel all over the world while updating their Instagram seemed to get on board with the ban plastic straws thing recently...
I've not personally tried the paper straws yet so to be fair I'm not completely averse to the idea, I just don't go to McDonalds very often anyway (which in itself is perhaps a good thing for the environment)! Out of all the things we could perhaps do to help the environment it doesn't seem like a big priority.
(*edit - not sure how the spell checker changed sea life into saline)