Someone else's grad student not interested in following up on a project
This is tricky for a couple of reasons. Adding a person as a co-author of a paper requires, at least, their agreement. Normally it also requires actual contribution to the work, but that idea is compromised in some fields, such as when a PI only contributes funds that make the research possible. Second, even giving an acknowledgement to a person normally requires their permission. This is more important in some fields than others. But if you attribute controversial statements to another you need their permission, generally.
However, there is nothing especially wrong about pushing someone a bit to elicit their permission or participation. This is especially true if you don't have a power (superior-inferior) relationship with them. Forcing subordinates is often done, though usually wrong. But that doesn't seem to be the relationship here.
I would send them each a note that you would like to pursue the topic and publish some results when done. Offer them co-authorship if they will participate effectively, but only acknowledgement of past contributions otherwise. Suggest that they talk it over with their own advisors. Then evaluate what you hear back.
I'm assuming above that all results so far are preliminary and tentative.
If there is nothing to do but the writing or minor clean-up this would be a different situation. In that case, write it up with all of you as co-authors and send it to them for approval or updating. Yes, you would be doing a large part of the work yourself, but that might be worth the effort.
The tricky thing here is that I don't know how much contribution each has made so far. Perhaps they are already due co-authorship of anything you produce. You will need to deal with that collectively.
If they have already contributed substantially, then a joint publication seems warranted (three co-authors). For that, all you need to do is get their permission. If they are unable to help work on the paper (no time, advisor objects, ...) then this might be your best option. Even a generous interpretation of substantially might be warranted.
I assume that, unless the topic is controversial, a publication is more valuable to them than to you, who already have a position.
To add to what Buffy said;
If you come up with a good idea that someone else further develops, you're entitled to some credit. In fact, you're entitled to a sincere invitation to a collaboration.
But if you had the idea but don't want to do any work on it, you're not entitled to a work-free co-authorship. You're also not entitled to keep squatting on the idea and obstruct others from doing anything with it.
So in your case, you should make it very clear to the students that they're welcome to get involved as co-authors, or to do nothing and get a nice acknowledgement.
How should I proceed?
- Do not continue to implore the uniniterested Ph.D. candidate to pursue the project. Whether it's a good idea for her to drop the project or not - it is her decision to make. You've mostly exhausted the extent to which it is legitimate to pressure them (except, perhaps, when you meet her face-to-face, you could ask her about it in the context of not having received a reply to your email; but again, don't press the matter too much).
- Do ask the uninterested Ph.D. candidate for permission to pursue the project further without her, including publishing papers, software, etc. She might not answer - in that case you're in the territory of a "contributor XYZ could not be reached" on future published work.
- Do remember to give appropriate credit to the uninterested Ph.D. candidate if and when you do publish something, even though she gave up the work. That may mean offering her co-authorship even though she "lazied out" on some of the effort, or an acknowledgement for her initial contribution - depending on the specifics.
- Do ask the interested Ph.D. candidate if there are unusual circumstances involving their uninterested colleague which be causing this situation (e.g. health issues); but
- Do not pry into the uninterested Ph.D. candidate's private life to figure out what's going on with her.