Truly Scrumptious's outfits were fashionable from about 1897 to 1907, and fashion changed rapidly after that. The car was retired in 1909, so the film is set in 1910 or later. If she was as rich as she appears to be, she would've worn more up-to-date fashions.
The opening sequence shows 4 races: the 1907 British Grand Prix, 1908 French Grand Prix, 1908 German Grand Prix, and 1909 British Grand Prix. The first British Grand Prix was in 1926. The German precursor to the Grand Prix was the Prinz-Heinrich-Fahrt in 1908; it became the Grand Prix in 1926.
When Grandpa Potts first arrives in Vulgaria, he says, "On behalf of Her Majesty, the Queen..." In 1910, the reigning British monarch was Edward VII, or George V, who took over after Edward died that same year.
Strictly speaking, the aircraft in the film is not a zeppelin, it's a semi-rigid airship. A zeppelin is vehicle with a rigid framework that contains multiple gas cells used to lift the craft into the air. Passenger compartments and mechanical areas are usually within the framework.
When the spies try to catch Chitty under a bridge, Mr. Scrumptious's car drives through their picture of the tunnel and into their truck. When car breaks through the picture, nothing is behind it. No truck. In the next cut, the car goes up the ramp and into the truck.
During the Baron's birthday party, Truly (as a doll) dances on top of a music box spindle within an unfolded screen of mirrors, visible on the floor for all the guests to see. When Caractacus (as a puppet) rises from his box and starts to dance, the camera pulls back, and Truly's music box is gone. When Caractacus goes over to sing a duet with Truly, she and her music box are visible again.
When the first Chitty crashes, the front end suspension is badly damaged and the left front wheel is bent. When the car arrives at the Potts workshop, the front end is not damaged and the steering works fine.
The two spies carry a painting of the bridge and tunnel, wrapped onto a pole about half the length of the tunnel's upper wall. The film jumps as it drops over the edge, and the pole suddenly spans the entire length of the wall.
When Mr. Potts pushes Chitty Chitty Bang Bang out of his workshop, the tires become coated with sand and dirt from the dirt driveway. In the next shot, the car is in the same place, and the tires are clean.
During the POSH song, Grandpa drops his pith helmet as the privy jerks downwards, and is nowhere in sight for the remainder of the scene, presumably dropping into the sea. It reappears when they land at the castle.
When Chitty Chitty Bang Bang is on the sand bar and Caractacus Potts begins telling his story, Jemima Potts holds a half-eaten apple. After the boat appears and heads towards them, only a few bites are missing from Jemima's apple.
When the Baron and Baroness sing and dance before attending the birthday party, the Baroness wears high-heeled shoes. She falls through the trapdoor wearing flat shoes. When she re-appears soon after, she's wearing high heels.
When the dog breaks free of its leash during "Toot Sweets", he knocks the pedal bike over and drags it across to the steps. Two shots later, when the pack of dogs runs toward the factory, they run up the steps where the bike is seen standing neatly upright, as before.
At the end of the seaside picnic, before Caractacus begins telling the pirate story, Truly ties the picnic basket onto the back of the car. The basket disappears as the car cruises on the water, then reappears when they drive out.
During the shots of the haircut and the man chasing Mr Potts, the carousel in the background is running at full speed during most of the shots. In one shot, it is slowing to a stop. About 15 seconds later, it's running at full speed again.
As the family are preparing the table for dinner, Mr Potts casts the tablecloth over the table, covering the children. In the next shot, camera positioned under the cloth, the children are standing right next to each other. When Mr Potts takes the cloth off, they are standing at least two feet away from each other.
Grandpa Potts complains that he will not get any sleep during the rainstorm due to his hut's leaky roof, yet during the part of the film where the hut is being transported, it is shown to be just a few feet square, with no room inside for Grandpa's bed; no bed is visible during his "posh" song, either; just a mostly-empty room.
In the seaside picnic scenes, the overhead shot of Chitty Chitty Bang Bang arriving at the beach and leaving the beach show an underwater ramp on the right side, used when the car was in the water. It's also visible during another beach scene.
During the main song sequence, Chitty Chitty Bang Bang passes a J94 "Austerity" Saddle Tank locomotive with "GWR" (Great Western Railway) on the side. The J94 was built in the 1940s, and the Great Western Railway never operated any J94s. The GWR livery was also deep green, not blue.
When they take Chitty out for the first time, Grandpa says he is going to Alaska. At the time, it was the District of Alaska or the Alaska Territory. It wouldn't be unusual for someone to shorten the name.
The junkyard owner tells the children that their father must buy the car before tomorrow morning, when the other man will come to pick it up. Counting the nights in the movie, the children go to sleep twice before Potts buys the car. He missed the deadline by one day.
The cooking machine heats plates hot enough to fry eggs and sausages before sending them down the ramp to the table. When the plates and metal roller frames arrive at the table, they should still be hot enough to cause burns. Mr. Potts and the children grab the plates with their bare hands.
When Truly runs into the pond, the car's engine is still running until Mr. Potts goes to get her. The car wasn't far enough in the pond to prevent her from backing out, like she did the first time she ended up in the pond after swerving to avoid the Potts children.
After Truly brings the children home, in mid-afternoon, their father tells them to go inside for tea. By the time they finish eating a bit later, their father tells the children to go up to bed because it's dark.
In the opening sequence, during the French Grand Prix, car 42 loses a wheel on a turn, followed by car 2A. There is an obvious jump in the editing where the first shot of car 42 losing its wheel was combined with the second shot of where it's been placed.
During the chase scene with Chitty as a boat, close-ups of the Baron and his officers on the bridge of his yacht show small wrinkles in the funnels behind them, revealing them as foil covered props. Wide shots show solid, brass-looking funnels.
The goof items below may give away important plot points.
At the end of the movie, Grandpa introduces Lord Scrumptious as "my Brigadier," the subject of the he told all of Caractacus' life. Lord Scrumptious probably wouldn't have changed his name since the time when Grandpa was his batman, so it shouldn't be a surprise to Grandpa that "His Brigadier" had lived nearby.
When the King's guards storm the main doors during the big fight, some boys throw several buckets of liquid on the stone to make the soldiers fall. More soldiers run in a few seconds later, and the stone is clean. In the following shot, the mess is back again.