Hi there, and thanks for coming to the forums with your question.
Without a pirate or navigator, you can only launch subs between outposts. So there's no way to force your sub to collide with the "queen hunter" sub of your opponent, except for a navigator or pirate of your own.
This is not a bug, but a smart combination of specialists by your opponent. He saved up his hires and combined them for a strategically *very* strong combo. His pirate will allow him to travel at 2x speed towards any of your subs, including a sub you're using to get your queen away.
Ways to counter this will likely include specialists. You might not have these (yet), which means you *could* simply be unlucky in this situation. Hopefully one of these will show up, and help you out of this situation.
* Hope to get a pirate, just 1 driller on the sub would be enough.
* Get a martyr to explode over his sub, which will be hard.
* Get a speed specialist (Lieutenant 1.5x speed, Helmsman 2x speed, Smuggler 3x speed) and outrun the "queen hunter", until you get a better hire or maybe attack his outposts, he might need to return / give up the chase.
* Get a Princess, if he kills your queen, the princess will turn into a queen. It's like an extra life.
* Get your Queen out of the sonar vision of your enemy. Subs don't give your enemy extra sonar vision, so he won't know where your queen is. He has to guess or attack an outpost blindly.
You see that there are (at least) 5 possible ways to counter this situation. It's an extremely difficult position to be in though! In my current game, someone is coming for my queen with a Navigator+Martyr, so if my opponent gets tired of the chase he can simply blow up one of my outposts... I'm glad he doesn't have a pirate though.
Good luck!