They couldn't even get NHS tracing app to work, think this will work? Wait till Dawn Butler, David Lammy and Clive whatshisface and the human rights lawyers get a look at the massive disparity in access to facilities/services that will skew heavily against their "BAME" wards/constituents/donors/supporters. The Cameroonian Tories are massively sensitive to any accusation of racism and refusal of the jab will LOOK like racism. Starmer may bumble into opposing vaccine passports. He'll oppose it for the wrong reasons (anti-racism against a race-neutral policy) but he'll get support for the right reasons from a lot of non-Labour people (liberals, libertarians, people concerned about human rights and freedom).
In short, yes, they want compliance passports/Nat. IDs but they may find the coalition of opposition too difficult to overcome, plus the Tory government is still ultra-sensitive about being portrayed as "the nasty party". Prepare for the worst, hope for the best and never stop opposing lockdown.