Mmm...I think there's a few people who seem to have the opinion that simply coming into Personal Union with a German state would be enough to annex a country to the HRE. I'm not sure why this idea is widespread, but I'm pretty sure it's nonsense. Even moreso, though, I think before throwing countries at the OP as an answer, you all first need to answer the question "why would England/Scotland/wherever actually want to join the HRE"? The problem here is that those countries have pretty much nothing to gain from joining, and they have a huge amount to lose - prestige, independence, and possibly even their royal titles. After all, if the HRE was really so happy-fun-joy to join, why didn't Hungary join it in the three hundred-odd years it was in Personal Union with Austria, or why didn't the UK join in the 120-odd years it was in Personal Union with the Electorate of Hanover? Simply because there was nothing to be gained in joining - and even more, because the Germans would likely to have rejected their entry. Outsider states weren't German and would have overpowered traditional German politics, issues and values. The Germans would be at risk of turning into second-rate citizens in their own country, especially considering how disparate and divided Germany was compared to the likes of the British states.
There was supposedly the offer to make the Elector of Saxony Holy Roman Emperor if he turned over Luther. Maybe if the Reformation started in Britain they could get the same offer?
With all due respect, this is a classic case of one-dimensional thinking. Pope Leo X didn't just offer Frederick the Imperial title (or that is, Papal funds to buy the title) because of throwing a random bribe at Frederick, he offered it because it suited his Papal ambitions in a number of different ways, and because it could actually have worked. Leo and a number of his predecessors wanted to detach the Hapsburgs from their sixty-ish year grip on the role and Frederick was genuinely a cultured, intelligent and most importantly popular major German ruler. He also was not powerful enough to continue to rule without Papal backing, at least initially. In other words, he was the kind of man who could have actually succeeded in winning the election - provided he had access to a pool of money the likes of the Pope's, in order to ensure the votes he needed - and once he did this, he would be largely a Papal puppet.
This offer would not have worked for anyone else. There were probably only a dozen men in Germany who were important enough to go in for the role and they were not the kind likely to win enough friends to be convincing of success. Outside of Germany there were several, but they were too powerful to be controlled by the Papacy afterwards, and importantly they were not German - the electors were very unlikely to go for a man who was not a German as they would fear outside interests. Most importantly, no-one else in Europe was sheltering Luther, and more than anything Leo X wanted Luther put on trial and executed for heresy, so he could nip the Protestant movement in the bud.
So simply, this offer to Frederick did not satisfy one Papal aim, it satisfied three - to rid Europe of Luther, to remove the Hapsburg control of the Imperial throne, and to turn the HREmperor into a Papal puppet, even if only for a while.