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, …)
I suppose you mean that you have filed a patent application in UK (not granted yet) and want to seek protection in US as well. If such is the scenario, the reply is yes, there is a way out for a UK patent application to be recognized in US. In order, to have a patent in US for the same invention (that was filed in UK) one has to file a patent application in US. If you wish to protect your invention in different countries, then you will have to submit a patent application in each of those countries.
There are two different filing options- a) Conventional filing route or filing in a foreign country. b) Filing a PCT application.
a) Conventional filing route or filing in a foreign country: In conventional filing there are two ways of filing a patent application- Firstly, filing a provisional application in any convention country i.e UK and file a complete application in each of the foreign countries one needs to seek protection( suppose US) within 12 months from the provisional being filed. Please note that the provisional application filed will help in securing a priority date. Secondly, filing a complete application in any of the convention country (to secure priority date) and further filing a complete application in each of the foreign countries in which you wish to protect your invention within 12 months from the priority date.
b) PCT route Filing a PCT application is a good strategy if you wish to protect your invention in more number of countries. By filing a PCT application, you get 30 months duration (sometimes even 31 months depending on the country) from the priority date to file patent applications in each country in which you wish to protect your invention. Suppose you have filed a patent application in UK on 12/12/14, then you have to file a PCT application within the twelve months from 12/12/14. It shall be note that a PCT application is always a complete application. The advantage of filing PCT application is that you will have 30 months duration (sometimes even 31 months depending on the country) from the priority date to file patent applications in each country in, which you wish to protect your invention (in your case say its US).
Please read the link provided for detail understanding of the aforementioned filing options http://www.invntree.com/blogs/what-are-different-patent-filing-options
Reply to the second question, there is no agreements between countries. The aforementioned route has to be followed to seek protection in different countries.