If I have an existing UK patent, is there a way to get it recognised in the US, or to convert it to be a US patent?
My guess is that there must be agreements between countries on how they handle patents, but is this true?
The usual way to have a patent recognized outside your home country¹ is to make a PCT application. A majority of countries are PCT members (Taiwan is the most notable exception). The PCT application must be made within 12 months of the original UK application (in the general case, there are exceptions). Usually, at this stage, you have a preliminary report from the UK office but no definitive grant or rejection.
The PCT application leads to a search of prior art and preliminary report on patentability. There is no such thing as a “worldwide patent”: the PCT application itself does not lead to a patent.
Within 30 months of the original filing in the UK (priority date), you may select countries¹ in which to deposit a national patent application, for example the US. The patent application is said to enter a national phase. The national patent office will have access to the international search report, but it is in no way bound by its recommendations. It can happen that one country accepts a patent and another denies it; you may get your US patent but not your UK patent or vice versa.
For more information, see the official PCT website.
Note that I am not a lawyer, I cannot guarantee that what I wrote above is true. The process is complicated; you will need a US attorney to handle the US patent application anyway (like in most countries). Your UK patent attorney should know how to advise you.
¹ Or group of countries (EPO in Europe, ARIPO in Africa, …)