Lineker's actually slightly wrong on this.
The language and actions are far more reminiscent of the British government in the 1930's than the Nazis. Although there was no specific legislation in place, the Home Secretary had the ultimate power to grant or deny entry. And the policy of appeasement meant that they didn't want to upset Germany by criticising its treatment of the German Jews. But nor did they want to help those Jews either.
There were strict conditions attached to any who wanted to come here and it was made clear that (a) they would have to be funded either by existing citizens or organisations (so not a penny of state funds) and (b) they were expected to assimilate as quickly as possible (e,g. strongly 'advised' not to speak German in public).
The right-wing press (i.e. most of it) used similarly inflammatory language in the context of the time, and this hardened public opinion.
There were around 500,000 German Jews at the starts of the 1930's. They could have easily been absorbed into the UK, US, Australia, South Africa, South America and Mandatory Palestine. Hitler, initially, just to get rid of them from Germany, rather than exterminating them. Even when it wa clear they were facing serious persecution, from the mid-1930's and even after Kristallnacht in 1938, doors remained closed.
In 1937, the ship St Louis, carried nearly 1,000 German and other mainly Jewish refugees to Cuba, who had promised entry visas. But unknown to them, the Cuban government revoked those while they were at sea. Many had applied for US visas and were planning only on using Cuba as a staging post. They were, bar a few, refused entry and the ship had to set sail. The next obvious port of call was the USA but they refused to let the ship dock, despite may being in the process of applying for visas. Eventually the ship returned to Europe and a few countries agreed to take a portion of the passengers. However over a quarter are known to have perished in the Holocaust.
