What did they stand to gain by fighting the Third Reich? What did they stand to lose from signing an armistice? I understand that the Third Reich had drawn up plans for an invasion of Britain, but would this still have gone through if London ended hostilities?
The public would have seen it as unacceptable, for one. Negotiations, as seen under Neville Chamberlain, clearly didn't work with Hitler. Another reason would have been that Britain's allies would have seen it as a huge betrayal to them (keep in mind that many of the governments-in-exile from Europe headquartered themselves in Britain).