> How does a digital ID solve an illegal immigration problem?
It doesn't. The kind of employer who would employ an illegal immigrant is certainly not going to ask to see ID of any kind. They would surely be especially wary of any electronic ID because that would make it easier to associate them with the immigrant. ID cards are only of any use to legal workers and honest employers.
If the UK wants the benefits of a solid ID it should look to Scandinavia. In Norway everyone has a unique number in the population register and this ID is your user ID for all state services. Employers can ask for this number and look you up. Of course it still doesn't prevent people working on the black for cash in hand but neither will an ID card or ID app.
There is no appetite for ID cards in Norway either, yet successive governments keep pushing the idea despite there being no compelling reason to believe that any problems will be solved by them.