Continuity mistake: Arthur ties a rope around the battlements and descends the tower's west wall. Tony then waves a torch to signal the soldiers to attack. He's standing at the same spot Arthur left from (the rope is still tied there), but in the long shot, his torch is waving from the tower's south wall instead of the west one. (00:41:15)
Factual error: The Vikings in this episode are fighting while wearing horned helmets. Contrary to popular belief, Vikings never wore horned helmets in battle. Some theories state the horned helmets were worn during special ceremonies, but never in battle or daily life.
Deliberate mistake: Here, as in many other episodes, Doug and Tony are time-shifted wearing period (in this case, Viking) clothing. But they tumble into their next adventure magically wearing their left-behind 20th Century clothes again. (00:45:30)
Suggested correction: A comment, not a correction to your astute and correct observation about the Time Tunnel. Aside from it being episodic television that allows characters to look the same in all situations, I believe Tony and Doug keep their clothing, hair length, lack of facial hair, etc, because the Time Tunnel "remembered" the exact disposition of Tony and Doug when they first entered the Tunnel, allowing them to keep their clothing, hair length, etc., as they slip through time. Any "Time Tunnel" aficionados wishing to add more to this would make a fascinating discussion.
A valid speculation with which I agree, but as the poster himself points out, not a correction.
Factual error: Wogan the Viking probably shouldn't laugh quite so enthusiastically. When he does, several of his very 20th Century gold fillings are visible. (00:09:30)
Revealing mistake: After Tony stabs Wogan with the sword, there is no blood on the blade when Tony pulls it out.
Continuity mistake: In the dungeon where Merlin reminds Doug and Tony they are from the future, Arthur's hands instantly move from by his sides to holding the collar of his tunic.
Continuity mistake: When Arthur has the standoff between himself and Doug and Tony, Doug is standing abreast of Tony in the frontal shots and slightly behind him in the views from the rear angles.
Revealing mistake: As Merlin circles the frozen Doug and Tony in the Time Tunnel, Tony blinks.
Other mistake: Usually Master Sergeant Jiggs and his men are posted as guards in the Time Tunnel control room at all times, but when Merlin shows up in the control room a second time, there are no guards to be seen.