He received his unusual first name in honour of Queen Anne, who was his sponsor at his baptism.
Poulett initially stood for election at Bridgwater in the 1768 general election, and although he was initially declared defeated the result was reversed on petition in 1769.
Throughout his career he was generally a supporter of Lord North, though occasionally voting against him.
In 1780 Poulett presented to St Mary's Church in Bridgwater a magnificent 17th-century painting of The Descent from the Cross, apparently captured when a Spanish warship was taken as a prize.
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