Please understand that internet trees for pre-1700 and pre-1500 are not sources. Many are wrong. Please look for primary sources or accredited secondary sources. Look for things like census, tax rolls, deeds, wills, church records, and court records. There are so many family histories written for this time period that were made up to make families look more connected to nobility that those are very suspect for the time period.
https://www.british-history.ac.uk/no-series/london-gazetteer-pre-fire/pp118-125 this site gives info on John Blackman and his land dealings. See section under 16th and 17th centuries.
https://www.british-history.ac.uk/vch/oxon/vol12/pp110-115 this mentions a Thomas Blackman "On the site of the Co-operative stores was a building once known as the old manor house, rebuilt in 1954. (fn. 78) In the 19th century its owners still received from the lords of the manor a rent charge of 5s., which in 1535 the abbey paid to Thomas Blackman and was later paid to his heirs. (fn. 79) The Blackmans were prominent in Eynsham from the later Middle Ages until c. 1600, frequently acting as manorial officials; in 1574 the family claimed arms. (fn. 80) The building, aligned with Mill Street and Abbey Street, was evidently an encroachment on the original square, and the residual payment and the tradition of manorial status suggest that it may have been the court house of the medieval manor. The Blackmans' successors by the mid 17th century were the Greens, and by then the building was the Green Dragon;"
https://www.british-history.ac.uk/london-hearth-tax/london-mddx/1666/stepney-ratcliffe-cross here is a Thomas Blackman on the Tax Rolls
https://www.british-history.ac.uk/search?query=Blackman&page=1 there are pages and pages here you can work through and perhaps find something that relates directly to your family. But please be very cautious of internet trees. Use them as clues not gospel. And work to verify anything you find with primary sources when at all possible.
https://discovery.nationalarchives.gov.uk/results/r?_q=Blackman&_dss=range&_ro=any&_p=1500 a number of Blackman listings to look at.
