The Telegraph has also run a similar story to HuffPo - again with quotes from the group representing Pakistani Christians in the UK, and the standard "We can't comment on individual cases" comment from the Home Office.
Reading between the lines I'd guess what's really going on is this - pressure group pushing for pre-emptive offer of no-strings asylum. Home Office saying "Sorry folks, that's not how the law works - if she gets on a plane here, and claims asylum on arrival, because that's how the law does work, her case will be considered in the same way as anyone else's. It may well be the case that she has a very strong case, but we can't pre-empt the legal process."
I very much doubt it's a case of "The UK will not offer her asylum."