As the second part of your question's been answered, I'll just answer the first part. Yes, both method and apparatus claims are both permissible for UK patents. Sometimes I write 'system' claims that are a bit of both!
You are also correct that methods of doing business, and programs for computers are both excluded - insofar as the invention relates to those things as such. The 'as such' is important - technical advances that happen to be implemented by software, and used in a business method, may still be patentable. The criterion is whether the contribution itself is solely in the excluded areas. The formal structure of the claim - method, apparatus, or whatever, is not important.
The leading case (at the moment!) is Aerotel/Macrossan, which provides a structured set of questions. There's also some practical guidance in the AT&T Knowledge Ventures case, called the AT&T signposts. The first chapter of the UK IPO patent manual http://www.ipo.gov.uk/practice-sec-001.pdf discusses all this.
It's easier appreciate after looking at some examples, rather than discussing it in the abstract. I'd recommend looking at some UK IPO decisions http://www.ipo.gov.uk/pro-types/pro-patent/pro-p-os/p-challenge-decision-results.htm . Go to the 'keyword or phrase' and filter by 'excluded field (allowed)' or 'excluded field (refused)' and have a look through the decisions.
What you include in the claim - and the description as a whole - can really make a difference. Emphasise the non-excluded technical aspects using the AT&T guidance.
It's worth noting that the EPO use different criteria (though the underlying law is supposed to be the same). I'd say they're more lenient. After an EP patent is granted, you then validate it in your chosen European countries, which includes the UK; this would be more expensive than just going direct through the UK IPO (and of course if you try and enforce it in the UK courts, they'll apply the UK excluded matter tests, and might revoke the patent!). But as for the US application you are considering, you can file the EP application within 12 months of your initial UK application, claiming priority.