Only thing I can think of now as a possibility as a
 scenario to put forward ?, is maybe one of Edmund's children or his brother or sister married a Quaker, if this Quaker 
had influence in the family of Edmund's affairs in Edmund's life
 to be able to authorise or arrange his (Edmund) funeral, that Quaker could have arranged for Edmund to be buried in a
 Quaker burial ground as Edmund being a none practicing Quaker of the Wear family.
It seems from what I've read  

, a 'practicing Quaker's' 
In Laws through marriage, of other beliefs in religion have a automatic right to be buried in a Quakers burial ground 
by association to a Quaker member. The 
none practicing Quaker of other religion denominations, could be, mother or father, or sibling brother or sister
 of the Quaker or mother and father
 In Law, brother or sister
 In Law  or the children of any aforesaid.