As far as I know (and I could be wrong), they use an electoral-college type deal – you don’t vote directly for the leader, but instead for your district’s MP in a FPTP vote, and then as you said the majority in parliament decides on the prime minister. They were trying recently to switch to proportional representation because it’s clearly better, but as far as I could tell it didn’t work.

So yes, just like in the US, her voting Green Party won’t change anything aside from strengthening the conservative party, unless she happens to be one of a couple of districts where the Greens have a competitive race. I actually voted third party in many elections in the US (in elections where Hitler wasn’t on the ballot), so I won’t say I think it’s throwing the vote away… I also however traded votes with family members a few times when one of us was in a competitive state and the other wasn’t, and I wanted them to vote third party while I was doing an establishment-party vote in a race where the outcome was more uncertain.