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According to this timeline, Theravada had the following significant times
http://www.buddhanet.net/e-learning/history/thera_timeline.htm
1899 • First Western Theravada monk (Gordon Douglas) ordains, in Burma.
1958 • Ven. Nyanaponika Thera establishes the Buddhist Publication Society in Sri Lanka to publish English-language books on Theravada Buddhism. • Sarvodaya Shramadana Movement is founded in Sri Lanka to bring Buddhist ideals to bear in solving pressing social problems. • Two Germans ordain at the Royal Thai Embassy in London, becoming the first to take full Theravada ordination in the West.
The Insight Meditation Society predates the first monastery
1970's • Refugees from war in Vietnam, Cambodia, and Laos settle in North America, Australia and Europe, establishing many Buddhist communities in the West. • Ven. Taungpulu Sayadaw and Dr. Rina Sircar, from Burma, establish the Taungpulu Kaba-Aye Monastery in Northern California, USA. • Ven. Ajahn Chah establishes Wat Pah Nanachat, a forest monastery in Thailand for training Western monks. • Insight Meditation Society, a lay meditation center, is founded in Massachusetts, USA. • Ven. Ajahn Chah travels to England to establish a small community of monks at the Hamsptead Vihara, which later moves to Sussex, England, now known as Chithurst Forest Monastery.
1980's • Lay meditation centers grow in popularity in North America, Australia and Europe. • First Theravada forest monastery in the USA (Bhavana Society) is established in West Virginia. • Amaravati Buddhist Monastery established in England by Ven. Ajahn Sumedho.
Depending on what your thoughts are on The Insight Meditation Society, you may consider that an earlier "monastery" and how strictly you interpret monastery.