Canada Day

Canada Day is the anniversary of the Confederation of Canada on 1 July, 1867.
http://canadaday.gc.ca/eng/1402068737686/1402077241673

On 1 July, 1867, Canada united to become a self-governing dominion of Great Britain with the confederation of four British Colonies: Nova Scotia; New Brunswick; Ontario; and Quebec.

Canada Day is celebrated as Canada’s Birthday.
http://www.statutoryholidays.com/canadaday.php

Australia Day

Australia Day is not celebrated as the anniversary of the Federation of Australia on 1 January, 1901.

On 1 January, 1901, Australia united to become a self-governing dominion of Great Britain with the Federation of the people of New South Wales, Victoria, South Australia, Queensland, Tasmania, and Western Australia. Australia Day is not celebrated as Australia’s Birthday, but as the anniversary of the arrival of the First Fleet of Convict ships from Great Britain on 26 January, 1788.

There can be no doubt that Australia Day, like Canada Day, should have always been held on the Nation’s Birthday. But, as the 1st of January is already a Public Holiday - it must fall to the day before:

"The Commonwealth of Australia, by contrast, did not evolve over the centuries. At the stroke of midnight on 31 December 1900 it sprang into existence in an instant, its Constitution with it, fully- armed like the goddess Pallas Athena.

S.E.K. Hulme
Samuel Griffith Society
http://www.samuelgriffith.org.au/papers/html/volume%201/chap2.htm

Like Canada , Australia has shuffled clumsily into a sort of independence in practice but a sharp lawyer could bring it all down.
" The Australia Act 1986 is the name given to a pair of laws: one an Act of the Commonwealth of Australia, the other an Act of the United Kingdom.. because of uncertainty as to which of the two parliaments had the ultimate authority to do so.

The Australia Act (Cth and UK) eliminated the remaining possibilities for the UK to legislate with effect in Australia, for the UK to be involved in Australian government, and for an appeal from any Australian court to a British court

Appeals to the Privy Council from decisions of the High Court were effectively ended by Acs of 1968 and the 1975. However, appeal remained from State courts, until 1986 .
The Commonwealth version refers to Australia as " an independent and federal nation". While this might be understood as a declaration of independence,., leaving open the question of when independence had been attained. There is no earlier declaration or grant of independence ".
The Oz constitution has not changed so technically , we're still under the Privy Council of British Lords and pollies. UK is out of the EU (almost) so we're all in the same wobbly mixup . Australia may ship its convicts and boat-people to the terra nullius of Yorkshire moors.

Yes, but I guess the question is whether or not you see January 26th as Australia's legal birthday, or Federation on JAN 1/DEC31 as Australia's legal birthday?

Easy. 29 April 1770 , Jimmy Cook gave it to George. "birthday" day of one's birth.

Yes, we could call it New Holland Day?

Yes , England was Dutch.
" In the Netherlands, Saxons occupied the territory south of the Frisians and north of the Franks. In the west it reached as far as the Gooi region, in the south as far as the Lower Rhine. The Saxon duchy of Hamaland played an important role in the formation of the duchy of Guelders.

The local language, although strongly influenced by standard Dutch, is still officially recognised as Dutch Low Saxon."

The Queen is Hanoverian German but as the Kaiser has left the building , then Holland's royals get the lot.