There were indeed people carting around collecting bodies during the plague! That sketch is supposedly inspired by the 1665 London plague outbreak chronicled by Daniel Dafoe, , where an unfortunate and (drunkenly) deeply sleeping piper was tossed into a corpse cart by "honest John Hayward" who went around town with a bell and a cart collecting bodies left out by people in the alleyways next to their houses. The piper is tossed into the cart with a few other corpses, and brought to the plague pit to be buried, where he wakes up, everyone panics a bit, and then the piper hops down and runs off. The piper is starving and dying anyways because it's so hard for him to get food or money; the piper was known to joke with people about the plague cart coming for him next week.

In his book, A Journal of the Plague Year, he actually goes into a a good bit of detail about Mr. Hayward, who is a Undersexton (undertaker) of a church. Hayward started by collecting dead in the usual manner, but as the plague progressed he began to go around his parish with a cart and collect corpses so that others wouldn't have to handle them. He also says that

"It is to be noted here that the dead-carts in the city were not confined to particular parishes, but one cart went through several parishes, according as the number of dead presented; nor were they tied to carry the dead to their respective parishes, but many of the dead taken up in the city were carried to the burying-ground in the out-parts for want of room."

The body collectors would go all around town, and then take the bodies where they could to be buried when the plague pits by their churches were full. This must have been backbreaking work - carting a heavy wooden wheelbarrow miles around town every day, then walking it far off to the outside of the city to be buried. These are your friends and neighbors too, and their bodies would be gruesome and smell horrible. The plague buriers were the most hardened people of them all, but even they would not venture into houses where everyone had died at once. They would ring bells, and wrap their faces with cloths soaked in vinegar to keep out the miasma they thought caused the plague. However, employment was scarce and times were hard, and people took jobs where they could find them. Burial was one of the few jobs left in town, and coaches (horse taxis, basically) were pressed into service to cart bodies out of town. He says that

"had it not been that the number of poor people who wanted employment and wanted bread ... necessity drove them to undertake anything and venture anything"

the bodies would never been cleared. The bodies were cleared at night, and Dafoe expresses amazement that the undertakers worked so hard. He has great respect for both the people clearing the bodies, most of whom died, and of the officials who had to employ them and kept the process going. However, he is disdainful about the handling of the crisis, noting that there were food shortages and insufficient warnings from the government when they knew what was happening. The poor were unable to work, and the ones that could stayed away from their homes and left food and money for their families on the doorstep so as to avoid spreading the disease. He mentions frequently that the living buried the dead, but that it could be very difficult, and that sometimes entire streets would be dead and the collectors would struggle to clear them. One of the biggest issues would be that no one was left to bury the dead, or even tell the body collectors that people were left to be dead. Sometimes streets and houses would be left dead for a good amount of time before someone would go to clear them.

Watchmen were posted at quarantined houses to make sure people didn't sneak out, and he describes an account of a watchman who was posted at a house he was near, who had stopped the dead-cart several times for the people inside, but had stopped hearing from the people inside. Eventually, the dead cart comes around and he bangs on the door and gets no response. Dafoe says that

"He continued knocking, and the bellman called out several times, 'Bring out your dead'; but nobody answered, till the man that drove the cart, being called to other houses, would stay no longer, and drove away.

The watchman gets a ladder and climbs up to the window to look, sees everyone is dead, and the dead-cart crew clears the house.

He also describes being stuck home in a quarantine and going rather stir crazy after a few weeks, so he decides to venture out of his house to the post office to send a letter to his brother. A purse of money has been dropped there, and no one will touch it. Finally, someone covers it in gunpowder, burns it, tosses the flaming purse in a bucket of water, and picks the coins out with tongs. Stir crazy people, after a month or so of isolation, also began to risk themselves to go to church and talk to people. When the plague started to end, people were so happy they flooded the streets and happily talked to random strangers.

So, the scene seems quite accurate, complete with the distinctive call of "bring out your dead!", and people tended to act the same then as they act now in quarantine. It's a fascinating book, and I highly recommend reading it if you're stuck at home with nothing to do. Many of the things he talks about are things that you can see in modern times, complete with greedy politicians, hoarding, and heartwarming accounts of selflessness in the face of death.