"The BOE study itself does not contain the word "infinitesimal"."
Check what I said, the author said it was " infinitesimal", The BOE study actually says "We find that the immigrant to native ratio has a small negative impact on average British wages." But dig deeper into the paper and the impacts are very small mostly to those in the unskilled sector and only if there 10 percentage point increase in immigrants working in that sector, its regional and it also states that it is short term. ( so yes it does account for impacts on different areas).
The migration advisory committee also found similar, I'd point you to the NEISR paper which explains in actual terms what the impact is, not falling pay ( because wages don't go down) but lower wage rises of about... 2 pence per week.
"The ONS did no study of the sort. Rather their data was used by MCLRG to create a set of rough estimations using a particular (flawed) model."
It was using ONS data, but I'll accept that. But still no evidence to the contrary provided, and this shows that immigrants haven't really had the suggested impact on houseprices. There have been far larger determinants.
"Nuffield states that immigration CAN reduce times in certain places, as long as the newbies are young and healthy and the natives leave the area."
The Nuffield study showed that it does reduce in most places. The "natives" don't leave, what happens is older people move out and are dispersed so lower waiting times. As far as I remember they discussed a lot of the "healthy migrant effect".
"Anyway, since drastically increasing the population with predominantly low-paid people in just a few years has barely any effect on anything at all, I patiently await your evidence of the real culprit."
Where to start? Automation, the massive reduction in trade union membership/availability in work places, the willingness of people to accept lower pay rates after 2008/9 in order to keep their jobs? There are lots of reasons, more than I care to list here.
Why did average wages rise between 2004-2008, even for the lowest paid in unskilled sectors, despite higher immigration?
But hey, I provided evidence that even when you take it as you do, proves that the claims about the impacts of immigration are massively over blown.