Frankly, it's a tough question because it depends on what you mean by "subjects". The short answer, however, is "No" if you mean formally so, but if you mean in practice did they answer to British authority, then yes they did. I think you're asking about the formality of it, though, and that's a "No".

The people of the Mandates were certainly subject to British authority. The Mandate gave the British "full powers of legislation and of administration, save as they may be limited by the terms of this mandate" (Article 1). This means that administration-wise, you're certainly correct. The territories were being held in "trust" of sorts, but the British ran them fully, subject to terms that they held a very large hand in crafting. Notably, the Mandate was handed to "His Britannic Majesty", but only to exercise it on behalf of the League of Nations.

This, again, is very different from being a member of the Commonwealth. In fact, it was made explicitly different. As Norman Bentwich noted in "Palestine Nationality and the Mandate" in 1939, a Palestine resident who came to England with a Palestine passport tried to argue he was there legally as a British subject. The courts rejected this, saying that Palestine was not transferred to the British, there is a separate nationality law that makes clear they were not British subjects, and a holder of a Palestine passport would not get the same visa-free treatment to enter countries that a British passport holder would.

Generally speaking, this is how all Mandates worked. They were not independent states, but neither were they colonies or giving the residents the full range of rights that the Mandatory Power held. So, in that sense, the residents were not subjects of the British King at all. The Permanent Mandates Commission of the League of Nations made this clear, saying that Mandatory inhabitants "are not invested with the nationality of the Mandatory power by reason of the protection extended to them."

Hopefully that answers your question!