Ostensibly, Labour was against immigration controls. This is evident from its opposition to the Commonwealth Immigrants Act 1968.
But your real question appears to be, how likely Labour would have passed the same law. I would argue that there's no great need for speculation.
Labour was voted into power during the 1964 election. Despite its earlier rhetoric, the party soon tightened administrative controls on immigration. By 1968, only six year after the earlier law, Labour enacted a blatantly racist immigration control, the Comnwealth Immigrants Act 1968. That law deprived Kenyan Asians of their right to enter the United Kingdom. Even though Labour presented it as geopolitical measure, the law was clearly drafted on racial grounds.
It would be undesirable in the report to speak in terms of coloured immigration;
instead the report should refer to Asian immigration.
- Cabinet Memorandum CAB 134/2640: Official Committee on Commonwealth Immigration
Although such memo were not made public until 1998, many contemporaries knew what the government was doing. For one thing, the fact that it exempted white Commonwealth citizens from the limitations, via a neat little loophole based on having British ancestors, was a dead give away. Although the bill enjoyed bipartisan support, it still provoked resistance from both Labour and Conservatives in Parliament.
My Lords, this suggests that, in spite of their discriminatory legislation, the Asian community were in no hurry to leave until the Government, led by Mr Duncan Sandys and Mr Enoch Powell strange bellwethers to lead and to stampede a Labour Xock panicked and threatened to break faith; and that is how the rush began. And as a result, we have this Bill, which The Times, a normally temperate newspaper describes as "probably the most shameful measure that Labour Members have ever been asked by their Whips to support."
- Speech of Baroness Asquith of Yarnbury in the House of Lords