The pedigree in Wake, based on the the Brudenell family documents, includes the generation of this William. [1]
The Brudenell pedigree in the visitation of Northamptonshire, attributed to Thomas, Lord Brudenell, and dated 1645, does not include this William. [2]
[1] Joan Wake, The Brudenells of Deene, Second edition (London: Cassell and Company Ltd., 1954), 484.
[2] Walter Charles Metcalfe, The visitations of Northamptonshire made in 1564 and 1618-19: with Northamptonshire pedigrees from various Harleian mss (London: Mitchell and Hughes, 1887), 168-171, [GoogleBooks].