Greetings! This is a rather interesting question, and it does reveal a fair bit about the "imperial connection" between the 'mother-country' of Britain and the "white-settler colonies" (or after the Balfour Declaration of 1926: dominions). Owing to OP's specific question (a good one at that), this response will be a bit briefer than previous discussions on the imperial nature and historical origins of the dominions. For a more in-depth look at the rights of the dominions prior to the 1931 Statute of Westminster, see this thread here. With that preamble out of the way, let's begin.
The Right of the Dominions
"The British prefer to emphasize the unity of the Empire, to them England and the Empire are one, and no such thing exists as an England conceived separately."
- A foreign observer on the British Empire in the postwar era
When London conceded the right to self-rule for New Zealand, Canada, Australia, and (after much difficulty and with a slightly different system), South Africa, it did so with the knowledge that there was a "Britannic connection" which would maintain the loyalty of the dominions to Whitehall back home. Ashley Jackson on this multi-faceted "link" which prompted the to-be dominion governments to remain "under" their British counterparts:
"these territories remained dependent upon Britain because Britain was responsible for their foreign affairs and defense, purchased the lion's share of their exports, supplied their imports, provided requisite inward investment, and held their sterling balances in London."
John Darwin, an imperial historian whose works on the British Empire I highly recommend as further reading, notes that such connections were the driving force behind the loyalty of the white-settler colonies before 1931:
"in two important respects they were being bound more closely to the old 'Mother-Country'. To compete in the global economy required heavy investment in the infrastructure of transport, and ever greater reliance on the shipping and sea-lanes that carried their products to Europe. Both drove them into a deeper sense of dependence on London and Liverpool, and sharpened the sense that their credit and capital were only as strong as their reputation in Britain. The colossal priority of economic development made it even less likely that they would cease to depend on British sea-power for strategic protection."
Thus, now that we have established the nature of the connections between Britain and the settlement colonies, we ought to turn to the opposite of dependence: autonomy.
In this regard, the elephant in the room was foreign policy. Until the Statute of Westminster, the key question of foreign policy remained firmly with London, and only on occasion (and after consultation with the mother country) could the settler colonies engage in affairs with neighbouring countries. In the Imperial Conference of 1911, when motioned to organise an Imperial Council to grant the dominions equal footing with Great Britain (a motion created by PM of New Zealand Joseph Ward), British PM Herbert Asquith responded with a clear stance on the whole matter:
"this would impair, if not altogether destroy, the authority of the Government of the United Kingdom in such grave matters as the conduct of foreign policy, the conclusion of treaties, the declaration of war, and, indeed, all those relations with foreign powers, necessarily of the most delicate character, which are now in the hands of the Imperial Government, subject to its responsibility and the Imperial Parliament. That authority cannot be shared." [italics as originally transcribed]
Frederick Sherwood Dunn, writing in 1927 (fittingly before the 1931 Statute) notes this legal condition of the dominions:
"From the viewpoint of international law, the Dominions were classed as autonomous colonies, not possessing international personality in consequence of the fact that their legal existence was bound up to that of the mother country. The Dominions might be permitted by Great Britain to engage in informal relations directly with other countries, but the ultimate responsibility in such matters rested entirely with the British government."
Of course, as a result of the Westminster Statute of 1931, the Dominions gained this freedom to conduct their own foreign affairs, set up embassies, and formally establish relations with various nations. Remember further that prior to this Imperial Conference, the Acts of Parliament in London could apply to the dominions, owing to the fact that the constitutional law of the dominions was (in theory) at the mercy of Whitehall. Autonomy from both points (foreign policy control and ultimate legislative control) were granted in the Statue:
"3.. It is hereby declared and enacted that the Parliament of a Dominion has full power to make laws having extra-territorial operation.
Thus whilst the dominions had transitioned from pre-First World War self-governing colonies to sovereign nations, they maintained loyalty to the mother country back in Europe (for the reasons outlined earlier). The "great liner" would finally sink after another World War exhausted Britain's ability to maintain the "imperial link" that had kept the dominions loyal and subservient prior to the 20th century.
Hope this helps, and feel free to ask any follow-ups as you see fit!