Sometimes with a public figure such as this they deposit their papers with an organisation they are associated with upon their death. I can't guarantee a family thing such as this "Parentalia" would be amongst academic or theological papers, but it is possible.
In his case, if he did this, it would probably be a church or university body he was associated with. You might try starting with the authorities at Chichester Cathedral. No idea if they would spare the time to look.
You may be aware that there is a John Burgon Society in the US. They appear to be Biblical literalists, more interested in theology, but they may know something about other papers.