Does language play any significant role in shaping national identity?

Language plays a huge role in shaping national identity, as any European knows. But from OP’s details, their question is really more, how does national identity get shaped in the absence of a distinct language? If it’s a sufficient but not a necessary condition, how do such countries get their own identity?

Let’s go shopping. Why does Austria have a distinct identity from Germany? Longstanding political separation, distinct history. Why does Cyprus have a distinct identity from Greece? To the extent it does at all (and the bicommunal experiment clearly failed): sense of grievance at Greece. Why does Belgium have a distinct identity? Distinct history again; and Belgium’s problem is it doesn’t have one identity, but three.

To speak of Australia (thank you Irene Colthurst!), there is a distinct Australian dialect, and it took only a generation to koineise out of the Sydney convicts’ dialects. But that dialect was not foregrounded in the formation of a national identity, which accelerated in the 1890s, in the leadup to federation. Australians then were mostly proud Britons (and remained so up until the 60s), to an extent that modern Australians find unintelligible. There was a republican narrative, but it was minority.

So what did it in Australia?

• Sheer distance, of course.
• The start of a distinct national mythology. Lots of romanticising of the Australian bush, and the virtues of its hard men. Poetry and short stories and paintings proved critical in the 1880s and 1890s.
• The popular notion that Australians were superior to the Britain British physically and, ultimately, socially. Australians were loyal Britons far too long, but they also took pride in defining themselves against the British.
• Sport. No, I don’t get it either. But sport. Both within Australia, and as an opportunity to beat the British at their own game. When Australia contemplated becoming a republic in 1999, one of the most pressing questions from the public was whether Australia could still participate in the Commonwealth Games. The international sporting competition where Australia is guaranteed to beat everyone else showing up. And which up until 1950 was called the Empire Games.
• The strong republican and Irish-Australian undercurrent: Australia was divided into English and Irish, with the English having the upper hand, but Catholicism had a strong presence, and bequeathed a legacy of resentment of the crown.
• Different, local bogeymen to what the Britain British had. It sounds horrible now, but the White Australia policy was really part of the how White Australians asserted their own identity: through pinpointing a regional threat that Mother Britain was indifferent about.
• The mythology that cemented Australian national identity was Gallipoli: the first time Australians fought in numbers overseas.
• It’s swept under the carpet now that they fought under the Union Jack, and that the reason returning Diggers found little resentment in Turkey was that, as far as the Turks were concerned, they had been English. In fact, it was much easier for ANZAC to be part of the national mythology, once the original diggers had died off.