As per my knowledge, there is no Branch of BBK or Bank of Bahrain & Kuwait in UAE (United Arab Emirates). It has got its Branches at:

Branches

There are 17 branches in the Kingdom of Bahrain

Adliya Financial Mall
Budaiya Financial Mall
Central Market
City Centre Branch
Diplomatic Area
Exhibition Road Financial Mall
Hamad Town Branch
Hidd Branch
Isa Town Financial Mall
Main Branch
Manama Branch (NRI Business)
Muharraq Financial Mall
Muharraq Suq Financial Mall
Refinery Branch
Riffa Financial Mall
Seef District Branch
Suq Wagif Branch

ATM

ATMs are available at all branches of BBK and the following locations

Al Aali Shopping Complex
Al-Reem Centre
Arad
Bahrain International Airport
Bahrain Mall
Batelco
BDF
City Centre
Directorate of Immigration and Passport
East Riffa
Gulf Air
Hamad Town
Jid Ali
Juffair
King Hamad University Hospital
Manama Plaza
Samaheej
Sanad
Seef Mall
Sh. Isa Avenue - near Abdulla Kamal Clinic

Bahrain Headquarter
Bank of Bahrain & Kuwait
43 Government Avenue , Manama , Kingdom of Bahrain
PO Box 597
Telephone
+973 17 22 33 88
Fax
+973 17 22 98 22
Cable
BAHKUBANK
Telex
8919
SWIFT
BBKUBHBM