Do you think you'd be equally diligent about teaching your daughter Slovak if you lived in the US?? (I ask because I know our future children would just NEVER learn Turkish if we lived in the US!) (I mean, I know you're in England right now, and we even will be soon, too, but I personally imagine that the US would make us EXTRA lazy!)
Honestly, I think it would make us even MORE diligent, for the reasons you mention. I agree that it would be so easy to just slide by with English, especially in an area without other Slovaks around. This time in England has showed us that at least for our family, no matter how much time Apo spends with K when he's home, it is not enough for her to keep anywhere near age level and we would have to step it up if we were staying long-term. And yet it is actually really important to both of us that K and any other children of ours would know both their heritage languages well. Really very important.
If we ever did move to the US permanently or semi-permanently, I think that I would take the plunge and go to minority language at home (CZ/SK at home, English outside) or even all CZ/SK, outside or in. I am unclear on whether this would involve me switching to speak actual Slovak. That would be less confusing in the sense that kids wouldn't have to decide whether to imitate Mama or Apo, but less workable in the sense that I would be mentally translating everything from Czech and second-guessing myself all the time. I could do that, but it would be more natural and spontaneous for me to speak Czech, which would turn us out some little Czechoslovak speakers - which, if we lived (semi-)permanently in the US, might not bother us that much.
I'm not sure how we would do it exactly, whether it would be truly all CZ/SK, all the time, or if I would move back and forth between that and English - since I wouldn't want to give up English interaction entirely - but I am fairly sure of one thing: we would be even more diligent than we are now!
Whether we would be SUCCESSFUL is of course another problem, but the desire is there.