Apparently these are the ancestors of David Cameron's puppeteer Gideon Osborne, who we never thought we'd miss, and they ran the Treasury for a century and a half in the Tudor and Stuart era.
But they were a bit nouveau then. There's no pedigree in the Essex Visitations or the Additional Pedigrees.
DNB only reports a tradition that they came from the north. Peter was at (not born at) Purleigh in 1440.
The internet name of his wife seems to be inspired by an Emma Bourchier who was a Countess of Essex.
It was probably their son who built Tyle Hall near Latchingdon, a moated Tudor manor house.
His son's 2nd son Peter moved to South Fambridge and got the Treasury job and made the family.
That part of Essex doesn't seem to be covered in VCH. I think it was dried-out bog without much history.
There was an Osborne family at Hartlip in Kent, but I'm not seeing any connection. They're done very sketchily in Hasted and mentioned in Berry's Kent pedigrees. Much later Hartlip descended through an heiress to a family called Tylden, and there's some sort of attempt going on to connect this name to Tyld Hall, but it isn't working.