I think it might be considered a case of duplicate content or mirror
website hostings even though on two different TLDs. Usually when search engines find duplicate content they choose one of the pages to list in the index, and then will ignore the other.
If you keep both, I think you'd need to sort out the hosting and the linking issue, having both domains on one hosting account would mean trouble.
You'd have to properly host the two
website hostings separately, on local hosts, and keep all the link sources separate. If you do this, I think they'd both stay in their local indexes. These would be two separate
website hostings on different local servers.
The other option which you can consider is to redirect the .com to the .co.uk.