The basis of America's claims are pathetic, and I've written about them before:
One of the claims the affidavit makes is that Ahmad sought to purchase 5000 tonnes of "sulfur / phosphate based fertilizers" from Pakistan, and it is alleged that he was trying to funnel this stuff to groups in Chechnya and Afghanistan. This is what the US generally calls free trade, but I bet one or two hairs are already standing up. Fertilizer equals explosives, right? Nope: ammonium nitrate based fertilizer equals explosives, sometimes. Sulphur or phosphate based fetiilizers equals vegetables. Further, to get 5000 tonnes of any substance at all into Afghanistan and Chechnya would require some considerable resources and logistical clout - which would be a great deal of effort just to grow pretty flowers in Kabul or Grozny. Ahmad was also accused of being in the possession of a "a several page tourist brochure of the Empire State Building" which, although dating back to 1973 when Ahmad's father visited the building, could entail a fiendish plan to detonate a major public building. The affidavit promises a specific reference to Mohammad Naeem Noor Khan in some of Babar Ahmad's 1997/8 correspondence, but never delivers it. What it does say is that someone asked a company to contact Ahmad in connection with the export of "mangoes" (scare quotes in original). Now, what various things can a Muslim man mean by "mangoes"? Presumably US intelligence imagines that there is some Quranic code in which "mangoes" is deciphered as "lots of weapons".
Ahmad is accused of mon-military support for mujahideen in Afghanistan or Chechnya during the late 1990s, but even if it is true, this was not illegal.
Haroon Rashid Aswat's case is even more strange. This man, alleged by two former American intelligence officials to have been an MI6 asset, is also blamed in the papers by unnamed "counterterrorism officials" of having been instrumental in the 7/7 attacks. He is said to have been to Afghanistan, trained in camps, and met bin Laden before being deported from Zambia to the United Kingdom in August last year. He is accused of having scoped out a potential terrorist training base in Oregon, the same surveillance operation attributed to unnamed Egyptian operatives under the instruction of Abu Hamza (Aswat was born in Yorkshire and is of Indian origin).
If there is evidence against these two individuals, then it would cause no difficulty at all to try them in the UK. Even supposing the two men are treated to a New York court rather than an offshore gulag, one assumes they will have stun belts to prevent any unfortunate outbursts.