Until the Extradition Act 2003 came into force at the beginning of 2004, extradition from Britain required the Government's consent.
Even if extradition was permitted by the courts in an individual case, it could still be blocked by the Home Secretary - as Jack Straw did in 2000 when he decided that the former Chilean dictator, Augusto Pinochet, should not be extradited to Spain on health grounds.
In a White Paper in 2001, Mr Straw argued that there was very little to be said for keeping the Home Secretary's discretion in cases involving extradition partners.
"As this is intended to be a fast-track system, it was thought important to reduce the number of stages a case goes through to a minimum," the White Paper explained.
Under the 2003 Act, the Home Secretary can block extradition only on very limited grounds - national security, for example, or if there are competing requests from different countries.
But it would not require a fresh Act of Parliament to stop the NatWest Three from being extradited. Mr Reid could simply make an order under the Act removing the United States from the list of countries to which fast-track extradition applies.
This would undoubtedly have diplomatic repercussions, but there is provision in the Act allowing people to be extradited under fast-track arrangements on a case-by-case basis.
In February, the High Court dismissed a claim that it was "unlawful and irrational" for Britain to keep the US on the list of countries designated for fast-track extradition under the 2003 Act.
Lawyers for Ian Norris, a businessman facing extradition to the US, had argued that regulations under the Act should meet Britain's international obligations.
Under the current extradition treaty with the US, signed in 1972, if Britain wants someone extradited from America it must produce enough evidence to satisfy the test of "probable cause".
The imbalance results from the Extradition Act 2003, which relieves the US of the obligation to establish a similar "prima facie" case when seeking the extradition of someone from Britain.
A new treaty with the US, signed in 2003, would no longer require the requesting state to produce evidence of wrongdoing, making it easier for Britain to have someone extradited from the US. But this treaty is not in force, because the US has so far failed to ratify it.
Last week, Alun Jones, QC, who acted for the NatWest Three, pointed out a further misfortune for his former clients: Britain is one of the few states willing to extradite its own nationals to the US.
"Even the EU instrument setting up the arrest warrant scheme permits a state to refuse extradition if it considers that the crime alleged was committed in its own territory, or where it has decided not to prosecute," Mr Jones added.