On July 25, 2019 the Most Rev. John Wilson (51), a priest and auxiliary bishop of Westminster was installed as the 11th Metropolitan Archbishop of Southwark. London is a unique city in that it contains two different ARCHdioceses, as well as also containing part of the territory of two dioceses. Everything south of the river Thames is in the Archdiocese of Southwark.
The archbishop employs the archdiocesan arms pictured below. They follow the more ancient custom in Britain of ensigning the shield with a bishop’s mitre rather than the more continental galero.
EXARANDORUM 
Up to me this is the archdiocesan coat of arms of Southwark. I havn’t seen Wilson’s c.o.a. yet.
You may be right. If so the archdiocese must have simply assumed them as the College of Arms won’t grant coats of arms to Catholic dioceses in England. Some of the previous archbishops of Southwark impaled their personal arms with a simple field argent with the cross of St. George (patron of the cathedral). However, most of the recent archbishops have borne their personal arms alone.
Small correction, Archbishop Wilson was a priest of the Diocese of Leeds before his appointment as an Auxiliary in Westminster.