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He is listed on page 52 of Watchmakers and Clockmakers of the World as Nathaniel Beasley, apprenticed in 1686 and free in the Clockmakers Company from 1694-1707.
He had entered a mark at Goldsmiths’ Hall around April 1697 as a free Clockmaker of Gunpowder Alley, New Street.
Land Tax assessment books are not well indexed but appear to list him in Three Legged Alley from 1703-13 and still in the Gunpowder Alley vicinity. He then went off.

I see that Nathaniel Baseley the son of Richard Baseley yeoman of Warwick County was apprenticed to Henry Hammond London clockmaker on the seventh day of March 1685 for seven years. His last name was indexed as Basoloy which I think goes along with secretary hand's "e's" to our eyes today.

James Desesar son of Abraham Desesar watch hand (sp?) maker of St Leonard, Shoreditch was apprenticed to Nathaniel Baseley London Clockmaker on June 10, 1698 for seven years. Baseley's name was interpreted by the person doing the indexing as Bastley, but it looks more like Basoloy to me.