Google don't look at Bing for their results
About a year ago Google complained that Bing were copying their results - http://searchengineland.com/google-bing-is-cheating-copying-our-search-results-62914 - but really that's an aside.
That said, in an ideal world I would also still recommend (wherever possible) that completely different content is created for UK and US markets.
Partially to dodge the duplicate content bullet (although with href lang it looks like this isn't going to be so much of an issue any more); but mainly because there are a lot of differences between UK and US consumers. Although we both speak English, the English we speak, the way we search, the messaging we respond to is different.
Obviously the option to create separate content isn't open to everyone (budgets, resources, etc). As such, if you can't stretch to creating separate content for each market I'd probably go with the hreflang implementation.
However it is important to note that all that really deals with is the duplicate content - if your pages aren't strong enough then they still won't rank, so you'll need to think about building links to these pages too.
I hope this helps,
Hannah