In Dickensian, Mrs. Gamp doesn't like the Midlands, as they are rampant with cracksmen, oysters and dippers.
I managed to find cracksman and dipper in Merriem-Webster, but who is an oyster?
I suggest that the transcriber of this episode misheard hoister†, which Partridge, A Dictionary of Slang and Unconventional English, 8th ed., 1984, p. 559 gives as a slang term from the 18th century into the 20th for a shoplifter.
† Or the actor playing Mrs. Gamp may have used a dialect in which the /h/ is dropped.
Some sort of a petty criminal, given (a) the next line is Well, rather a few petty criminals than a murderer on the loose and (b) that oyster is placed in a list of petty criminals in between cracksman (burglar or safe-cracker) and a dipper (pickpocket), definitions found in Merriam-Webster.
The Oxford English Dictionary (OED) has no record of oyster as any sort of petty criminal.
In the seven volume Slang and Its Analogues Past and Present, there is also no entry for oyster that refers to a petty criminal.
It could be an erroneous or one-off use. If Dickens had actually used it to mean a criminal it seems one or both of the above resources would list it as such.
Or you could write the BBC and/or the authors of the script and ask them what they meant.