There is a good possiblity that Abrahams may have acquired the diary/diaries, possibly from an estate clearing, perused them and later sold them. He has been known to sell archival documents and around the time of NB's death. NB's collection of old maps/books would surely have caught Abrahams interest if these were sold.
Found this transaction by Abrahams in National Archives of England :
SOCIETY OF ANTIQUARIES. Transcripts, notes and extracts by early officers and Fellows of SA conc. antiquities, pictures, MSS, etc., some derived from Stukeley's book, SAL/MS/265, with added information SAL/MS/264 1718-mid 18th cent., with later additions
Paper; ff. iv + 226. 1718-mid 18th century, with later additions. Folio. Half leather, brown, with marbled boards. Miscellaneous London notes, fol. iv.
Archival history: ... Sold by Aleck Abrahams to SA, April 1912 (fol. ii).
http://www.nationalarchives.gov.uk/a2a/records.aspx?cat=118-sal_1&cid=-1#-1NB's interest in following Mrs. S, must have been for her connection with John Bunyan. I bet he had a copy of Pilgrim's Progress and would have loved the opportunity to view the author's portrait.