And chamberlain included Churchill in his war cabinet - and then when Churchill became prime minister he included chamberlain in his war cabinet as Lord President of the Council where Neville ran the group in churchill’s absence. A mixed and famously unfortunate history but chamberlain’s citizenship and loyalty to UK was not a matter of serious question. ( later Churchill critiqued chamberlain for, as you said, having been one of those who failed to, or who were opposed to, preparing for the war ).
Everybody puts almost the entire blame on Chamberlain for not taking the Nazis seriously and the Munich debacle. But to be fair there were many others including the two PMs before Chamberlain, Ramsey McDonald and Stanley Baldwin, who pretty much ignored Churchill’s warnings.
Although he was excluded from the cabinet and any decision-making, Churchill had over the years built up a very large network of friends, diplomats, and people in high places who had the info he needed about the Nazis. There was no Freedom Of Information Act in Britain, and Churchill's informers risked being thrown into prison for giving Churchill top secret info.