Can we at least move away from discussing it with the same discompassionate and deliberately exclusionary/dehumanising language that the right wing use to stoke up nationalistic differences between people?
No human's existence is illegal. The people being discussed are undocumented migrants.
Given the way the Home Office, by policy, treats even documented migrants I think we can all agree it's fully rational for people to be concerned about coming forwards to engage with the health system over here. The health system itself has sadly become more exclusionary by Tory policy in recent years, although had some successes in fighting to retain anonymity of some migrant health data that Home Office demanded. But as a system is also not good/idle (particularly GP level) accommodating those with no fixed abode (for whatever reason, including rough sleeping and undocumented migration).
Reaching people who are rationally concerned about coming forward to register for a vaccine is going to be important for everyone. Destigmatising their existence, including in how we refer to them, is one of the first steps.