To clarify my own understanding, which may be wrong, but is based on our own use of DMCA takedowns that we've issued over the years....
Your host does not to do anything if it is UK based, as DMCA solely applies to US law, hence US hosts. If the site is not hosted in the US, it needn't be taken down by hosts, the responsibility passes to the site owner.
That said, you don't want to ignore it. A judgement against you could lead to seizure of any US assets or, maybe more worrying, stop you from ever being able to visit the US. There may also be treaties that allow US judgements to be enforced through UK courts, not sure.
The big problem with either a DMCA takedown, or a counter notice is that the other party might try to sue for damages/losses caused by the notice or counter-notice. The case would then be handled in a US court where, even if you win, you quite probably have to pay all your own lawyers fees.
Even if you just roll over and comply, I'm not sure if that implies an admission of guilt.
If I were on the receiving end of a DMCA I'd Seriously consider getting some professional legal advice, from somebody expert in US law. A much cheaper alternative would be to establish some kind of dialogue with the complainant, if you feel confident that you can do so without incriminating yourself...
I suspect it would also be a good idea to remove the content while any dialogue is taking place, else istr the complainant might argue that you 'knowingly' infringed even after being notified.
As stated, I am not certain of above facts, somebody else may have more accurate information to offer?