The data actually implies UK citizens have been consistently moving out of London. The absolute white British population in 1971 was 6,500,000 but was only 3,239,281 in 2021 [1].

Of course not all UK citizens are white British, and the negative fertility rate would've applied downward pressure, and it's conceivable that white Londoners emigrated to Australia etc at a higher rate than other parts of the UK. But none of these factors can really explain a 50% decline in population in the same way that internal migration can. It certainly becomes very difficult to reconcile the numbers if any signifiant number of people from predominantly white areas had been moving into London.

I agree with you that most of the change in average UK house prices is driven by London. In a zero immigration scenario it seems likely that London prices would be radically lower with the rest of the country having much less, if any, change.

[1] https://en.wikipedia.org/wiki/Ethnic_groups_in_London

Yes, London house prices would likely be lower. But we'd also be poorer on average, since immigrants are a net benefit to the economy. Moreover the combination of a starkly aging population with higher costs of labour would cause a real strain on the public purse, like Japan is seeing. On balance I much prefer the current situation. Though I guess I would say that, having an immigrant parent.