Once the millions of migrants in Germany have the requisite EU papers they would be able to flood into the UK. Shengen is a complete red herring.
They do:
> You can travel to the so-called Schengen states (Belgium, Germany, Denmark, Estonia, Finland, France, Greece, Ireland, Iceland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands, Norway, Austria, Poland, Portugal, Romania , Sweden, Switzerland, Slovakia, Slovenia, Spain, Czech Republic, Hungary) without a visa, if you do not stay longer than three months and you do not work there. [0]
In the EU that's a moot point anyway as I haven't seen any border control in years. So at least travel-wise they're quite free to move.
[0] https://handbookgermany.de/en/rights-laws/asylum/blue-passpo...
you need a visa, is not free to move.