So I've been looking into it since I got such wonderful positive feedback, and it turns out Ivan and Liz were on pretty good terms, and that Ivan was annoyed Elizabeth focused more on commerce then searching for a military alliance with the Tsardom of Russia. He even proposed to her in his later life, so he could seek asylum in case his rule was overthrown. It was Ivan's son, Feodor, that jeopardised English-Russian relations, by removing exclusive trading rights and ejecting ambassadors. Elizabeth even offered Feodor a military alliance in an attempt to reinstate relations. With the addition of an incredibly Orthodox Elizabeth the 1st instead of an incredibly Protestant one, I would say it's quite plausible for this to work. As soon as my exams are over, I'm gonna try and make a timeline of this!