When did the accusations of convict stain against white Australians stop being culturally significant?

TL;DR:

In the 1950s it was still a stain and mark of derision not to be spoken of. In the 1960s attitudes changes. By the 1970s it had become a badge of pride which it still is for some today.

Long Answer:

I'll quote extensively from two interesting articles on the topic and so just abbreviate which one the quote is from. One article itself quotes extensively, so in those cases I'll list the source.

1. The Conversation, June 8, 2015, "Stain or badge of honour. Convict heritage inspires mixed feelings" (TC)
2. The Australian, March 28, 2008, "Taking stock of our birthstain" (TA)

NOTE: 2. may be hidden behind paywall or popup, but should be fully readable as one-off

Overall, one in five Australian have convict heritage, in Tasmania that number is an astonishing 74% (TC).

It was a topic not to spoken about.

As late as the 1960s, Tasmania’s Library Board refused permission to doctoral student Peter Bolger to publish convict names for fear of embarrassing their descendants. (TC)

The last convict died in 1938! (TC)

Attitudes began to change in the 1960s but there was resistance.

In the 1950s and ‘60s, historians argued that Australians should not romanticise either the convict system or the people within it.

In a footnote in Mind-forg'd manacles it's commented that as late as 1999, historian

[...] A. G. L. Shaw restated his claim that he believed convicts to be a disreputable lot belonging to a criminal class and that Botany Bay was settled for penal rather than strategic reasons. See A.G.L. Shaw, "The Convict Question," Tasmanian Historical Studies 6, 2 (1999): 4-16.

And Australian historian Mollie Gillen still maintained in 1985 in The search for John Small, First Fleeter (1985) that they were all

… raggle-taggle nobodies … who walked the streets as idle and profligate persons. (TC)

Also

At the bicentenary celebrations in 1988, it was perfectly in keeping with the Australian tradition of avoiding the convicts that the only speaker who referred to them directly was the Prince of Wales. (TA)

but the article goes on the mention that this was also because of the overall change in attitudes which saw the First Fleet as an invasion with disastrous consequences for indigenous Australians.

However in private people started to become extremely interested years before. The records were never destroyed and became public in 1951. Interestingly the US TV series Roots may have had something to do with it

The trickle of family researchers that began in the late 1960s turned into a flood during the '70s, and by the '80s Australians regularly descended on the archives in droves. Initially motivated by the 1970 bicentenary of Captain Cook's mapping of eastern Australia, they had been further intrigued about family antecedents by the popular television series created for the American bicentenary in 1976 called Roots. (TA)

Those who dove into the research were shocked to find that it wasn't merely government policy to hide "the stain", but that it was their own ancestors

Until family historians began researching in numbers in the '70s, the extent to which convict history had been covered up at an individual level was unknown. Sometimes it originated with the convicts, sometimes it was created by their descendants. Regardless of where it began, by the early decades of the 20th century the screen erected to protect families that were founded by convicts had become genuine amnesia. (TA)

It was the emancipation from the British Empire in two World Wars that also had a large impact on the change, as Australia started to develop a national pride and consciousness

Whether standing in the dock, sailing into the unknown or facing the lash, laughter - often in the form of cheek - was their best, indeed their only, defence. Any number of Australia's criminal founders literally laughed in the face of death. So, with equal courage, did their heirs in two world wars. Their humour, laced with bravado, dry, self-deprecating, face-saving, is still a defining characteristic of the society they left behind. (TA)

Nowadays, the convict stain is (ab)used by those who don't really have any idea about public attitude about it. Such as in 2007 when Taj Din al-Hilali said of Muslim Australians

seeking the ultimate insult but misjudging its impact in modern Australia, [...] "We came as free people. We bought our own tickets. We are more entitled to Australia than they are." (TA)

The reaction

was one of incredulous laughter. Despite the sheik's angst, he and other Australians of the Islamic faith will in due course discover, as so many immigrants before them have done, that humour, including the ability to laugh at yourself, is the passport to acceptance in Australian society. (TA)

PS: for an academic treatment of the topic you can also read Convict ancestry: a neglected aspect of Australian identity, but I haven't yet had the time to do so.