The change in the royal title came about following the Imperial Conference of 1926, when the various governments of the Empire agreed that "the group of self-governing communities composed of Great Britain and the Dominions" were "autonomous Communities ... equal in status". The Conference declined to pursue developing any kind of Imperial Constitution, preferring instead to draw out a few specific consequences of that agreement. In the Balfour Declaration, section IV(a) asks for legislation to change the King's title. There is also a longer discussion in V(a) about treaties, which affects the examples that you give in the question.
Both of your 1930 examples follow the recommendations in this Declaration.
- The convention on conflict of nationality laws is entered into by the King, with the governments of his realms separately represented and enumerated. It lists two plenipotentiaries "For Great Britain and Northern Ireland". The omission of "the United Kingdom of" is not material, much like talking about "the Kingdom of Sweden" versus just "Sweden".
- The agreement on Bulgarian financial obligations is between the Government of the United Kingdom of Great Britain and Northern Ireland, and various other governments.
They both refer to Northern Ireland rather than Ireland but differ in form because they use the different structures of a treaty between sovereigns and an agreement between governments. As noted in the Declaration, prior usage was not always obvious to interpret, because of confusion about whether (say) Canada was meant to be bound by a treaty entered into in the name of the King of Great Britain. Likewise, R v. Jameson  2 QB 425 is an important case on extent of legislation to British possessions - in the past, it was not always explicit whether an Act was for the UK only, and this had to be deduced from context.
In debate on the 1927 Act in the Commons, the Home Secretary Sir William Joynson-Hicks (as he then was) said:
It is clear that it is a misnomer to continue to talk of "The United Kingdom of Great Britain and Ireland." Quite obviously, since Southern Ireland has been granted Dominion Home Rule, there is no such thing as "The United Kingdom of Great Britain and Ireland," but there is still "The United Kingdom of Great Britain and Northern Ireland." Consequently, after full consultation and interviews with the representatives of Northern Ireland, with the Government of Northern Ireland, it has been decided to ask the House to alter the title of Parliament- and to make it "The Parliament of the United Kingdom of Great Britain and Northern Ireland."
Certain portions of Ireland have, as it were, slipped out of the United Kingdom, and we now call it the United Kingdom of Great Britain and Northern Ireland, but it is still the United Kingdom, and this is still the Thirty-fourth Parliament of the United Kingdom.
It is clear that he - on behalf of the government - saw the Act as sufficient to change anything already defined in statute which needed to be changed. (It was also debated whether the Great Seal needed to be renamed as well, etc.) The change to the name of the country did not need statutory authorization in contexts where the old name was not required by statute.
- Diplomatic formulae are not statutory and so aren't mentioned here; the royal prerogative is already enough for the treaty language to be as envisaged going forward.
- British passports also changed in 1927 to using "United Kingdom of
Great Britain and Northern Ireland" in place of "and Ireland". At the time, passport issuing was entirely on the basis of the royal prerogative, without any legislative regime.
- Various other official publications, like statistical reports, used the new name going forward.
Some other provisions in the Declaration were given effect in the Statute of Westminster 1931. This should be seen as part of a continuing effort to let the legal situation follow the policy (Commonwealth nations are equal autonomous peers) which itself followed the facts on the ground (the UK government could no longer expect to rule them directly). So there was not precisely a "flag day" when the UK's name changed, but there was a process of recognizing that the name ought to change, some wrangling over how, and the fairly rapid implementation of the change in 1927, at least on the part of the UK government itself.