Bedivere at the time wasn't a Servant. And in a less nitpicky note, he wasn't a part of a Holy Grail War, he was concealing his identity for completely different reasons.
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Well, before this chapter I was thinking one of those "time travel but not really" things, like having somehow Rayshifted into a Singularity or some kind of alternate pruned timeline. But since this chapter establishes Tsukumo really is his grandmother from a linear timeline leading to his birth (or, at least, her well-being is somehow intrinsically tied to his)... I don't know. Current FGO/Extella lore Re: parallel worlds and timelines is just a confusing mess to me.