Hey Alex,
That is tough - after reading the title of your post I assumed this would be a quick "add hreflang tags and set international targeting in Google Search Console." How long have the hreflang tags and international targeting been set up? The answer may be just to 'wait'.
Do you have offices/locations in both the US and UK? If so, you could set up Google Business pages for each and link to the appropriate site given the region. This would also allow you to build local citations and/or business listings using country specific directories/websites and make sure those go to the appropriate site. Example:
Submit UK Site to Yell.com ("The UK's leading online business directory")
Submit US Site to YellowPages.com ("YP is a leading local marketing solutions provider in the US")