According to census records, Anstey was probably born at Otterton (Devon) in 1843. (See particularly the 1881 Census; Otterton is also mentioned in 1851, 1871 and 1911; the 1901 Census says "Atterton", but that's probably a transcription error; the 1891 Census says "East Budleigh", which is a neighbouring village.) The estimated birth year varies between 1841 and 1844, but perhaps he was the Henry Anstey whose birth was registered at St. Thomas (Devon) in the second quarter of 1843.
Some other links I was sent on HCA,
His last will:
Anstey with family: