Britain has not competed in the football finals since Rome 1960, and last took part in the qualifiers in 1971 when they failed to reach the 1972 Munich Games.
Professional players have been allowed to take part in the Games since 1984, but forming a Team GB has always been blocked by the Scottish, Welsh and Northern Ireland football associations, amid fears that they might lose their independent status within FIFA.
However, when London was awarded the Games a British team was obliged to compete as hosts.
"I hope it is not a one-off from what I have experienced so far," British captain Giggs said at one of the first media conferences held at Olympic Park.
"Of course I am Welsh and proud of it, but for me to have the chance of playing in the biggest tournament in the world right now, it has been great and I hope it continues."
Asked if he would lobby Welsh officials to rethink their past policy of opposing a British team, the 38-year-old replied: "Whether people have been English, Welsh, whatever, those I have met have wished me well and the team well.
"I hope fellow British footballers can experience the same thing in the future. I am not sure I would lobby anyone, I am not a politician but I hope we carry on in the future."
Craig Bellamy, 33, a Welshman like Giggs, said the experience was totally different to the one he was used to in the Premier League.
"It is new to us, we are meeting other athletes, eating with them, being part of a huge British squad," he said.
"It feels very different from just being in a football team, it feels very special being with other athletes all trying to win medals for Britain.
"We are kept away a bit as Premier League players usually, but this has been so different and we are embracing it.
"I personally never thought I would ever be involved in anything like this and it is something to cherish for the rest of my life."
Coach Stuart Pearce's squad face Brazil in a friendly at Middlesbrough on Friday - the first British Olympic football match for 41 years - and he said the squad were getting stronger after training in Spain.
"We had a great week and gelled very quickly," he said.
"We are some way behind the others in terms of preparation - for example I think the United Arab Emirates have played 17 friendlies, but all this squad know each other well and we have hit the ground running and have no injuries after the training game against Mexico."
Britain face the UAE, Uruguay and Senegal in their group matches, with their opening game against Senegal at Old Trafford on July 26, the day before the Games officially open.