I came across a picture of Bishop Challoner with a full wig recently which I tweeted, never before having seen a picture of a Catholic bishop wearing a wig and a mitre.
A book on the practice of Catholicism in London in 1805 (written in 1905) gives a bit more information:
"The law of shaving was in force, but before the French Revolution the custom was so common that it was not particularly distinctive of the clergy; and when, later on, it became fashionable to let whiskers grow, the priests frequently wore what was called the "clerical inch", which equally prevented any distinctive appearance of their ecclesiastical state. This was in order to save them from being insulted in the streets-no doubt a useful precaution in the eighteenth century, but one which rapidly became unnecessary in the nineteenth. It is curious to note that Dr Milner always dressed in a brown coat and was not recognised as a priest, while the first to adopt a stricter attitude was Rev Joseph Berington, who was a well-known writer of doubtful orthodoxy: he began to dress regularly in black early in the nineteenth century, and was blamed by many for so doing. I need not say that the Roman collar was unknown till some thirty years later. Wigs were getting less common a century ago, though the older clergy still wore them. Dr Douglass is always represented with one of a close-fitting type, quite different from that which Dr Chailoner used to wear; but Dr Poynter never wore a wig, nor did Dr Milner. They would have powdered their hair instead, and this custom of powdering was strictly observed by all those who ministered at the altar almost till Wiseman's time. Dr Weathers, Bishop Auxiliary to Cardinal Manning, who was ordained priest in 1838, is reported to have been the first to discard the custom of powdering before singing Mass."