I'm not getting the Britain & Austria vs Prussia & Russia conflict. Why would Prussia and Russia both attack Austria and why would Britain get involved in the conflict?
If there's a obvious change about to happen in the European balance of power then that could concern Britain enough to take action, but that's what happened OTL.
If Russia looks like it's becoming more powerful or if there's a close alliance with Prussia then maybe. I think Britain would prefer to build a larger coalition first.
Rather than Britain choosing to go to war, Prussia could provoke a war by attacking Hanover. Prussia might be confident of winning battles on the ground, but it would still hold back from from war with Britain. Britain wouldn't care how long it took, but it would ensure that Prussia was eventually beaten and Hanover regained.