BBC2 did it for a few years and I think it ended on the BBC after Euro 96, ITV did it for France 1998 & Euro 2004. Christian O'Connell did World Cuppa on ITV4 in 2006 and James Corden's World Cup Live on ITV in 2010 which must have been the last tournament comedy crossover programme.
There was also Johnny Vaughan's World Cup Extra in 2002, shown on BBC Choice at 9pm every night and then repeated around 11pm on BBC1.
As you say, Fantasy Football began on BBC2 in 1994 and ran for three series, with specials during Euro 96 their final outing on the BBC. Then in 1998 it was on ITV, who poached it for a huge amount of money, and showed it live several times a week (the Euro 96 shows were just once a week). I remember David Baddiel saying if England made the final they'd go and just find somewhere to do the show - "If England get to the World Cup Final, I don't suppose anyone will be bothered half an hour of telly is a bit rubbish."
As you say, it was then revived for Euro 2004. No Jeff Astle this time, as he'd died, and I'm not sure this run was particularly well-remembered, it was still good though. I remember Des Lynam was doing a diary of the tournament, his last as a presenter of course. in the Telegraph at the time and in it he complained about having to stop for trailers for it, saying "ITV are desperate to prop up this ailing show". The miserable git.