We have been 'letting people in' as you quaintly put it for hundreds of years. I'm not sure that Mr Blair can claim all the credit can he?
Didn't a 'Mr Powell' make a now infamous speech about the dangers of 'letting people in' yet you seem intent on giving the credit to just one man - how so?
Well ignoring periods of empire and the slave trade, it's also one of those cases where people confuse the net value, as governments often do too, with the rate of immigration. The trends were well in place by the time Maggie departed office and Major took over. As the economy had gone through its eddies and recovered, the emigration rate slowed and immigration rate stabilised and began to increase. Take the population growth since, better education (including English skills across the board), years of uneven European boom, the end of the Cold War, cheaper and faster travel, and you are in the present.
Sure Tony and Gordon had their moments of hubris, ignoring expert advice re transitional controls in the EU as an example and actions of our European partners, but to lay it all on them is silly and lazy. And this is not looking at non-EU immigration and the Commonwealth specifically, which came in waves and coincided with our transition to a post-Imperial period and is still going strong.
Pull and push factors remain complex, and far beyond the control of just one charismatic PM; in general: more advanced economies will always hoover up more people; ditto for global connections and service-oriented output.
The 'problem' isn't 'letting people in' but ensuring inequality is minimised, integration efforts are supported and the benefits of immigration to the country aren't left to trickle down glacially, from one class to the next. Something we've been singularly terrible at, in systematic terms, and a major downside of Thatcherism; Tony took it on, but didn't finish and got sidetracked by Iraq and his flirtation with Bush. Then the post-crash climate took over.