Where does a United Kingdom citizen that does business with an American citizen United Arab Emirates take their matter to in case of disputes?

R u talking about entity(business) or the case of individual?

If in case of Individual, it will be UAE as the issues comes under the territorial jurisdiction of the same.

In case of business entity, it depends in case there is a Arbitration clause is there, it can be any where U.K. or U.S. or U.A.E. or any other country also, they can mutually appoint the Arbitrator and can be resolved as per the International Arbitration Covenants and the same is adopted in India's Arbitration & Conciliation Act,1994. I hope it will be same there also with regard to international Arbitration is concern. Thanks.