Is there evidence you become immune after catching it?
None whatever. Any speculation about immunity and how long it lasts, or herd immunity and what percentage is needed, is just that - speculation and guesswork, and not particularly educated guesswork at that, when you consider how many of the worlds virologists and epidemiologists are slackjawed at what the UK is doing. There is a certain amount of weak anecdotal evidence to the contrary, i.e. that you can in fact get it more than once, and if you do that subsequent infections have more serious outcomes.
I find it disturbing how many scientists and medical professionals are repeating this nonsense (for example Clare Gerada, former Chair of the RCGP, said last night that she was just getting over it and was glad she would now be able to go back to frontline clinical work knowing that she was immune). Anyone with a medical training does know really that this is at best an optimistic guess and a high risk strategy. I think it's a real-time example of how insidious propaganda is, and how readily people who should know better go along with the flock. I suspect that that, rather than a specific desire to euthanise the old and ill, is the sociopathic experiment Cummings and the so-called 'behavioural unit' are engaged in. They must be delighted with such an early and unexpected opportunity to refine all the strategies they developed during the Brexit project.