The appointment of a single agency marks a change in strategy for UCI, which had previously used a roster of separate agencies - including the latter two on the pitchlist - on a project basis.
It is also a major win for Brazen, which was set up by Wheeler last July, with co-director Emma Thornley, who also left Mason Williams to set up the firm.
The success of films such as Harry Potter and the Philosopher's Stone and Lord of the Rings have propelled cinema audiences - once in decline - to their highest for 30 years.
The challenge for Brazen is to make UCI stand out from its competitors - particularly as it introduces new luxury cinemas with more spacious seating and bar facilities.
As well as day-to-day media relations and crisis management, Brazen is to launch campaigns on groups such as students and families.
Cinemas have traditionally held most appeal for the 18-to-24 age group but, according to Wheeler, it has proved harder to entice older customers.
A six-strong team at Brazen will deal with the account, reporting to UCI comms manager Vicki Locke and marketing director Janet Smith.
UCI - owned by Viacom subsidiary Paramount Pictures and French media giant Vivendi Universal - is the UK's second biggest cinema group behind the market leading Odeon chain, with 39 cinemas in the UK and Ireland.
However, it is the market leader in the multiplex cinema category.
It was reported this week that Vivendi CEO Jean-Marie Messier was looking to sell his company's stake in UCI.