To add more context, this quotation is his response to the situation he's chosen to consider in this section of his essay. "Whenever I go into a foreign country or a prison or any similar place they always ask me what is my religion". The difficulty here is not about his beliefs, it's about how he can accurately answer that question when posed by some official or other.
So, he's not changing his beliefs, he's changing the label he uses when describing his beliefs to different audiences.
Russell assumes that to a philosophical audience the word "atheist" means "a person whose beliefs entail the non-existence of God", whereas "agnostic" means "a person who thinks there is no conclusive argument for or against the existence of God" (he explicitly states here that he thinks there is no conclusive argument against God. He has let it go without saying that he thinks there's no conclusive argument for God either).
Therefore, to a philosophical audience he uses the word "agnostic" to describe his beliefs. He can't present a proof of God's non-existence from his own priors, which is what a "pure philosophical audience" would understand by the word "atheist".
Then, Russell assumes that to the ordinary man in the street, "atheist" means "a person who does not believe in God". By this definition, Russell is an atheist, since "God exists" is not among his beliefs.
Basically he's advocating talking to people in their own language rather than getting too hung up on the proper philosophical definitions at all times. Therefore that there's no contradiction in describing himself as an agnostic in one context and an atheist in another. In particular he sees no contradiction in publicly describing himself as an atheist and nevertheless believing that one cannot prove the non-existence of God.
By raising the Olympian Gods as an example I think he is also teasing those who want to draw what he believes to be an overly-pedantic distinction between agnosticism and atheism. He asserts (rightly or wrongly) that there are many things other than the Christian God in which we don't believe, and which if absolutely pressed we would acknowledge can't be disproved either, but we don't go around in everyday life saying that we're agnostic on the subject.
Is he implying that because there are so many religions that they all
must not be true?
No, he's saying that because he describes himself in everyday language as not believing in the Olympian Gods, and because (he says) his disbelief in those Gods is of a kind with his disbelief in the Christian God and every hypothetical God, that he should also describe himself in everyday language as an atheist. He's not describing anyone who disbelieves in the Olympian Gods and believes in the Christian God, and he's not saying that just because a person disbelieves in the Olympian Gods then they should be an atheist.