Assumption - you are taking about (normal) utility patents that cover the actual functioning of an invention.
There are time limits for filing in follow-on countries. In the straightforward way of filing in the UK after first filing in the US, the time limit is one year. Since it generally takes over a year to have a US patent granted it may already be too late for them to file anywhere else. On the other hand they might already have an application pending in the UK.
And there is another way of getting a UK patent by filing a PCT application within a year of the US filing. (I am assuming the first filing was in the US.)
A PCT filing delays the choice of countries and could lead to a decision to get serious about a UK patent by entering the national stage in the UK up to 30 months after the first filing or by entering the regional phase at the EPO at the 30 month point. An EPO patent can then lead to a UK patent.
You should first double check that there really is a granted US patent. Many people who post here about a “patent” are actually looking at a publication of a patent application, not a granted patent. If the number starts with something that looks a year, it’s an application. This does not mean there isn’t also a granted patent but, if so, it is a different document. When you look it up you can (if in google patents) click on Global Dossier and see where else it might be filed.
An important consideration is that there may be a patent in the UK held by a completely different patent owner that covers some aspect of your planned product. Imagine you are planning to make an electric lawnmower but some patent on a manual lawnmower blade is infringed by your product. People can have a freedom-to-operate search done to dig these up.
Regarding getting exclusive rights in the UK - I think you know that since you didn’t invent the product you can’t patent it. And patents that are already granted or even published are prior art to anyone elsewhere in the world who happens to independently invent and tries to get protection. They can’t because it isn’t new to the world. I do not know of any way to get exclusivity than a patent but being local first-mover, good brand name and powerful distribution partners can help.
I do not particularly recommend this, but disclosing the patent number most likely will get you a much better answer here because we could look up its history.
