To focus on Paget's case however, He was accused of criminal conversion as a preliminary to divorce proceedings [around May 1809]. The subsequent divorce proceedings were heard in the Consistory Court of the Bishop of London (took about a month for the final decree to come through) [June-July 1809].

Just for the record, this was standard practice (but a "crim. con." is actually a "criminal conversation"). The crim. con. (i.e., adultery) was usually the first stage of the divorce proceeding, and had to have a named defendant, i.e., a husband didn't sue his wife for adultery in general, but sued another man for damages for adultery with his wife (in civil court).

The first official proceeding in a judicial separation (which preceded a divorce) was held before the London Consistory Court, better known as the Doctors' Commons (named for the building in which the court sat).

Anyway, the purpose of a crim. con. civil case was to establish the wife's guilt, and the verdict could be presented as evidence in the following proceeding in the Doctors' Commons which would establish the separation.

There is a detailed explanation of the divorce procedure in Stella Tillyard's Aristocrats when she describes Sarah Lennox's divorce from her first husband, Charles Bunbury (or rather, his divorce from her, since she could not instigate the proceedings).

The Duke of Wellington was reputed to be the target of crim. con. suits in his later career.