You seem to know what really happened. I read somewhere that Thompson's command of English was "limited" and that created shooting problems. However, when I bought this collection, I found his English quite fluid. (I don't find his accent especially thick either, but then again, I am also Argentine, so what do I know?)
No matter what, as you know, in the last 3 episodes Thompson's character was replaced, or --even more pathetic-- in the very last episode he is reduced to a few lines and the ingrate role of a bumbling idle useless womanizer. I felt bad about it. I prefer to consider this series as consisting on only (the first) ten episodes, as I don't really find reasonable any explanation given for his sacking.
As you probably are aware, Thompson later went to have a good cinema career in Germany (where he married Lilly Palmer until her death) so at some point he evidently spoke good German (he was of German-Swiss descent).