As for Sweden.... well that could be interesting. I think the main problems were the fact that Sweden's throne was elective, so there was a perception that she was marrying down in station, and the fact that Erik was a King. That was the problem with several of her Protestant suitors. Elizabeth wouldn't marry a reigning King. It just wouldn't be feasible. But a marriage between Elizabeth and one of Erik's brothers, John or Charles would be interesting.
On the contrary, all of the problems relating to the elective nature of the Swedish throne, as it had been constitutionally established to be in the Code of the King in Magnus (IV) Eriksson's Law of the Land from the 1350s and re-affirmed in King Christopher's Law of the Land in the 15th century had been resolved with the introduction of hereditary monarchy by Gustav (I) Vasa at the Riksdag of Västerås in 1544.
But of course, both John and Charles were far more politically savvy men than Eric, as demonstrated by that both of them managed to usurp the throne at later points. Seeing Charles' usurpation was directly related to the fact that he took control of the Realm Council while his nephew Sigismund was away in Poland, I don't think that an Anglo-Swedish personal union will last too long. Sooner or later either John or Charles will rebel and seize the crown.
What I am curious about though is how the Swedish Church would develop. It is interesting to note that for much of the 16th century that was a very unclear matter. John III, married to a Polish Catholic Princess, was very keen on a reconciliation with Rome, and as such, under him the Church of Sweden was very reminiscent of the Anglican Church as a kind of
via media compromise between Catholicism and Protestantism, while Charles IX was very attracted to Calvinist doctrine. It was first at the end of the 16th century that Lutheranism was firmly established as the official religion of the Swedish state.