Further non-UK examples:
Norway means "northern way".
East London, South Africa (double hit!)
Palmerston North, New Zealand is much larger than the original Palmerston which is on the South Island.
Northern Mariana Islands
Northwest Territories, Canada
Western Australia (NT and SA also kind of count, someone already mentioned NSW)
Western Sahara (a partially recognised state)
West Bank - while there are a few sources for "East Bank" it is now the country of Jordan.
Western Samoa renamed itself to just Samoa in 1997. The eastern islands of that group are American Samoa.
Two groups of countries use a common currency with a compass point name: the West African CFA franc and the East Caribbean dollar, with no opposite directional currency. (There is the Central African CFA franc as well as the previous and proposed East African shilling.)
Shaanxi, Yunnan, Jiangxi provinces of China, as well as Tibet which is Xizang in Chinese. There are 8 provinces which are counterpart compass points of each other (Shandong/xi, Guangdong/xi, Hebei/nan and Hubei/nan).
Hokkaido, Japan.
Beijing and Nanjing, China and Tokyo, Japan mean N, S and E capitals, albeit Tokyo refers to being east of the earlier Japanese capital at Kyoto, not the Chinese capital. There is currently no W capital, but Xi'an in China has been called that in the past.
Yugoslavia
Yuzhno-Sakhalinsk, Russia
Severnaya Zemlya, Russia