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At the end of The War Games, the Second Doctor (Patrick Troughton) was last seen spinning away into the void, crying, "Stop! You're making me giddy!"
The next aired episode, Spearhead From Space, at the beginning of the next season, showed the Third Doctor (Jon Pertwee, wearing his predecessor's clothes) fall out of the Tardis, unconscious.
Why was no regeneration sequence filmed? Had the Third Doctor not been cast at the end of the filming of "The War Games"?
This is a tough one, because it's very hard to prove a negative, but in the 48 years since this happened, we have not received any (well documented) conclusive answer to this question.
To first answer your sub-question/postulation.
The final episode of The War Games aired June 21st, 1969. Jon Pertwee was announced as the Third Doctor 4 days before on June 17th, 1969. So it is likely that they had cast the Third Doctor in time for filming a regeneration.
So why didn't they? Well, there are several theories. However, one thing you should keep in mind is this is only the second regeneration ever. There was no established pattern yet, in fact compared to Classic-Who, NuWho is remarkable uniform in how they handle regenerations.
At the time, Troughton was not as well liked as he may be now. The show was still in its infancy and changing the main actor something of a gimmick. The fade-to-black could have been to leave open the possibility of ending the show there with a clear resolution. The Doctor was exiled, the end. However, this theory doesn't hold much water since The War Games was written largely to get the doctor to earth to make production cheaper the following season.
As Daniel Rosman mentioned in the comments to the question:
According to the editors of the Discontinuity Guide, it's because there's a whole unfilmed season between the end of The War Games and the beginning of Spearhead, during which the Second Doctor was forced to work for the Celestial Intervention Agency. (The theory is based on some very tenuous bits of evidence to do with the Doctor's claimed age and a reuse of an actor from the trial.)
I also think this is unlikely as Patrick Troughton himself said that he felt 3 years as The Doctor was enough and left the show. Since they knew they were filming the last season, and were trying to set up a cheaper production the following season, it doesn't make sense to have another planned, if unfilmed, season in space with the actor that already announced his departure.
The Second Doctor's regeneration was shown in the comic story The Night Walkers. Published in November 1969, it is widely considered to be the conclusion of Season 6B. While it doesn't solve the continuity gap, it did happen in close enough proximity (IRL time) to the final episode of The War Games to be pre-planned. Remember, regeneration didn't have an established pattern yet, and Doctor Who had been publishing comic stories in TV Comic for upwards of 5 years at this point. This could have been an attempt to connect the two forms of media. But that is mere speculation.
So to wrap that all up, we simply don't know. Personally, I'm not happy with this answer since so much of it is speculation. But I think that is, in a way, reflective of the state of facts surrounding this situation.
Well technically considering the fifth to sixth doctor regeneration is very similar with swirling faces
And the second doctor had swirling in his regeneration
We may have met The Master earlier than we expected
