the "Great British public." is a light-hearted image of what the British public should be like (stereotypically British and willing to do/share eccentric, heart-warming or funny events). "Great" plays upon both "British" and "public".
It also has slight overtones of pride in being British.
I agree with PaulQ on what it might mean, and often does. But it's impossible to be sure what particular nuance (if any) is being suggested here without context. What exactly is this the title of? Can you tell us something about this article or publication, or whatever it is?
By the way, please do put your entire question in the title. The title should simply reflect the phrase or usage you are asking about. This will prevent the thread from coming up in searches on the basis of the words in your title that are not relevant.
I don't associate the phrase with sarcasm (post 2); not quite pride (post 4) for me but affection. But maybe there's not a lot to choose between them. As Matching Mole says, the meaning will depend on the context.