Two other things happened in "The Invasion" that would be much more important later. Nicholas Courtney again appeared as Lethbridge-Stewart of U.N.I.T., but this time his character had been promoted to the rank of Brigadier, a role he played with each subsequent Doctor. And John Levene, originally hired to play a Cyberman, instead ended up replacing another actor to play a new character, Corporal Benton, who returns in later seasons as Sergeant Benton.
The reconstructed episodes give you an idea of the plot, but I find them very difficult to watch, and in fact didn't watch them all. Surviving clips were interspersed with still photographs, voiceovers (some audio tracks survived when the film didn't), and occasional scrolling messages to tell you what action had just happened. I ended up just watching episode 3, which is complete.
Only the human villains are seen in episode three, and boy, are they creepy. Tobias Vaughn is the arrogant and ambitious rich industrialist who's hooked up with the Cybermen; his sidekick, Packer, has a tiny brain and a taste for revenge and sadism. Zoe, alas, isn't in this episode. But I got to watch the Doctor and Jamie. Their body language is like puppies tumbling from problem to problem, totally into each other's space and most of the time holding on to each other. Jamie does a bit more of this than the Doctor; he's constantly grabbing the Doctor by the sleeve or the lapel to get his attention, which makes sense when the Doctor is often in his own mental world about twenty steps ahead and doesn't want to be bothered with Jamie's constant questions. But there's fondness in it, too. The Doctor often grasps Jamie by the shoulder, seemingly to reassure him, just as often as he simply grabs Jamie and pulls him where he wants him to go. They're two parts of a gestalt.
Also, just that one episode is a good examplar of the Second Doctor's character. He keeps silent and polite and seemingly harmless with Tobias Vaughn, saying things like "You're too kind," while waiting for his chance to get what he wants. He's afraid for the safety of Jamie and Zoe and the guest star Isabelle, yet at the same time confident that he can outwit Vaughn easily with his superior brain; and he does outwit Vaughn, mostly because Vaughn depends on Packer to do his work for him, and Packer fails. Vaughn chastises Packer for his failings; the Doctor encourages Jamie to think things through, whether Jamie arrives at the same answer as the Doctor or not.
My favorite line was Jamie to the Doctor: "You know something? You're a clever wee chappie!"
Transcript of episode. Information on The Invasion. And a bit more, with credits.