Hi Spencer,
The rel="alternate" hreflang="x" tag inclusion along with the Geotargeting option in Google Webmaster Tools (if the websites are not ccTLDs but generic domains) are solutions for these situations: When you have two websites targeting different countries with the same language and they're competing with each other in SERPs, cannibalizing their opportunities and that might end up also with content duplication issues at some point.
About your concern: "Currently for some of our keywords US is #1, UK is #4. If we implement rel alternate, will it just remove our US page".
Your US website would be removed for the UK search result pages indeed and although in the short term this might seem as counterproductive, the reality is that this result from a relevancy perspective doesn't belong there since you already have your UK website targeting those users especially.
This also means that you can also be losing opportunity by having UK users going to the US version that is not as relevant for them, so their experience can be poorer and the conversions might not be the same. Additionally, as I mentioned before you might end-up suffering from content cannibalization / duplication issues with both websites showing the same or very similar content without giving signals to Google that they're not really duplicates but two versions targeting different audiences.
Because the previous in the long-run the best is that you make the most out of your international Web versions (country versions in this case) and give the necessary signals to Google so these start ranking in these countries search results instead.
In case your US version is a .com domain and it's also attracting rankings, organic traffic and conversions from abroad, excluding the UK (for which you have your own UK website version) then you can leave that website without a specific geolocalization in Google Webmaster Tools and just specifying the language in the hreflang attribute (not the country) but if you have a specific website version for the UK then you should definitely specify that this website (whether it's a .uk ccTLD, subdirectory, subdomain or another generic domain) is geotargeting to this country.
I hope this helps!