The original speech was given to the House of Commons, which didn’t record any speeches. If you’re saying that the radio speech was live, of course it was.
I don’t think its impossible for an actor to imitate Churchill.
Clearly, this Norman Shelley guy was pretty convincing.
As for the other posts, they are conflicting.
Obviously, Churchill read the speech more than once with this Shelley guy reading it at least once himself.
Who read the speech that is in the Youtube clip in the above link?
I don’t think any of us saying “sounds like Churchill to me” is of much weight - our idea of what Churchill sounded like comes from all these recordings!
As Exapno Mapcase has pointed out all the speeches were originally given on the House of Commons and clearly not recorded, they were subsequently broadcast on the BBC and - whatever Irving says - there is little doubt it was Churchill who delivered them. There are too many contemporary accounts of his doing so to be in much doubt, many of them by Conservatives with little love for Churchill. The difficulty is knowing if the recordings we have all heard were made at the time.
This is complicated because it is clear that Shelley did record at least one of the speeches later in the war and Churchill re-recorded some of the them in the early 50’s for the BBC - quite possibly they had lost the war-time originals!
So in answer to the OP: Churchill did deliver the speeches but whether that YouTube recording was of him making it in June 1940; who knows.