For later events, the climax of The War Games seems clear: 'The time has come for you to change your appearance and begin your exile.' Later continuity (including the events of The Five Doctors and The Two Doctors) makes this progression almost impossible.
The War Games is clearly the Doctor's first contact with the Time Lords since he left Gallifrey. If he was in any sort of contact with them (such as that shown in The Two Doctors) he could not have avoided being caught. The possibility that one of the Time Lords at the Doctor's trial is Chancellor Goth of The Deadly Assassin (they're played by the same actor) opens up the possibility of CIA involvement. The Celestial Intervention Agency, mentioned in The Deadly Assassin, are the Time Lords who directed many of the Doctor's wanderings, particularly during his fourth incarnation. During the trial in The War Games the three Time Lords, referred to as 'the Tribunal' and not the High Council in Terror of the Autons, tacitly admit that the Doctor has 'a part to play' in the battle against evil.
Let us suppose that, at the end of The War Games, far from being cast into exile, as those Gallifreyans watching (including the High council) would assume, the Doctor is being given his freedom, at a price. The second Doctor that we see in The Two Doctors is clearly not the same as he was before The War Games. He seems much older, and his TARDIS has dual control, the Time Lords monitoring his every move.
In short, in return for his freedom, he's intervening at the behest of the CIA. He is given a Stattenheim remote control device, a 'privilege' which he 'earned'. The sixth Doctor reacts with amazement, always having wanted one (cf The Mark of the Rani), indicating that some, if not all, of his memories of this time were later removed. (However, season 6(b) would seem not to be the reason why the sixth Doctor doesn't know anything to his advantage in The Two Doctors, as the drug administered to the second Doctor affects memory.)
Sarah's discovery of the second Doctor's recorder in the wooden control room ('the old one') in The Masque of Mandragora indicates that this was used during this period. The trouble with this control room, the Doctor tells her, was that he was never in control of where the TARDIS landed.
At some point during his travels alone the Doctor takes time off to visit the Brigadier at a UNIT reunion, which would have been impossible with the pre-The War Games unsteerable TARDIS (The Five Doctors). This bends rather than breaks the (legal rather than physical) laws of time. Thus he is aware of Jamie and Zoe's return to their own people (The War Games).
When talking to the Brigadier he mentions creatures that were 'covered with hair, used to hop like kangaroos'. This sounds very like the description of the Medusoids (Frontier in Space), who the Doctor was detained by while on the way to a peace conference. Going anywhere deliberately again points at post season six adventures. After many years of him working alone (see Pyramids of Mars) the CIA concluded that the Doctor functioned better with companions, and allowed him to recover Jamie (the Highlands in the aftermath of the '45 rebellion was hardly a safe place, with the Duke of Cumberland's forces massacring the Scots) and Victoria (from her unlikely adoption). The fact that, in The Two Doctors, the Doctor has dropped her off to learn graphology indicates that he's got every faith in being able to pick her up again.
Later, the CIA do actually enforce the Doctor's sentence and exile him to Earth, the Tribunal continuing to oversee him. (Perhaps the CIA's schemes were uncovered by the High Council, or alternatively the Doctor started to rebel against them, disconnecting the Mark III Emergency Transceiver from the console in order to stop getting messages from them (The Creature from the Pit).)
The CIA knew that the Doctor's exile would coincide with a spate of alien attacks on Earth. There might have been a small judicial hearing (the Doctor in The Trial of a Time Lord states that he's been involved in several such enquiries before). At the start of Spearhead from Space, the newly regenerated Doctor is found wearing a ring, a bracelet and a TARDIS homing watch, none of which he had at the end of The War Games. In Frontier in Space, knowing the Master is listening, the Doctor says that the High Council tried him. (His version of events is very different from that seen on screen.)