UEFA selected Rome's Stadio Olimpico to be the venue for the opening game of the 24-team tournament, despite interest from Glasgow's Hampden Park.
Speaking at a press conference in Nyon, which was broadcast on its website, UEFA president Aleksander Ceferin said the decision to take the games from Brussels was unanimous after the Belgian authorities failed to provide guarantees the new stadium would be ready.
That likely means that if England qualifies for Euro 2020, they'd be placed into Group D and would have at least two games in London.
The company contracted to build the stadium in the northern suburb of Grimbergen, BTP Ghelamco, said they were able to build the 60,000-capacity Eurostadium in 18 months, but the Flanders region in which it will be situated has yet to issue a building permit.
In a statement, FA chief executive Martin Glenn said: "We are delighted to help UEFA by hosting the additional four matches at Wembley Stadium". They didn't expect a decision until January and they did not know whether that would be a yes or a no. Experts from [our] administration felt it was a high risk for UEFA to wait.
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The competition is set to be played under a new format with fixtures taking place in 12 different nations as a one-off event to celebrate the tournament's 16th anniversary.
Group A games will be played in Rome and Baku, while Group B games will take place in Saint Petersburg and Copenhagen. Wales has never staged a Euro or World Cup final and this was its one and only chance of doing so.
It had been hoped that construction would get under way by 2016, but so far the stadium is yet to be granted planning permission.
Bilbao and Dublin have been paired together in Group E and Munich and Budapest will host Group F games.
While the decision has come as a blow to the Welsh, and Belgians, it is a major coup for the Football Association, London and Wembley Stadium, which hosted several games in the 1996 tournament.
UEFA also divided host cities into pairs, with each pair sharing matches in a given group.