Factual error: Dr. Johnson completed his dictionary in 1755, 7 years before the Prince Regent was born (1762). So it's unlikely for him to be asking the Prince for patronage.
Factual error: During the theatre play you can see they are using electric stage lights. Nice move for a series plotted almost 300 years ago. (00:01:55)
Factual error: At Ms. Miggins' when she says, "There's ever so much hard work," you can see her wearing a bra under her corset. Bras didn't exist at that time. (00:06:35)
Factual error: After the Shadow has left Blackadder a captive, she says how much she hates squirrels, and fires two shots from a single shot pistol.
Factual error: When Blackadder has disguised himself as the Prince and is speaking to Wellington, Wellington says, "I have with me here the most recent briefs from my General in the field". Blackadder, deliberately misunderstanding, replies "If you would like to pop them in the laundry basket". Blackadder is referring to the short, tight style of men's modern underwear, known as briefs, which were not invented until the 1930s.
Answer: His first name is "Sod Off". Blackadder asks him in series 3 where this came from, and he replies "Well, when I was a kid, living in the gutter, I'd go up to the other kids and say 'Hello, I'm Baldrick'. And they'd say, 'Yes, we know. Sod off, Baldrick.'"