I am presenting a paper at a computer science conference.
The paper I submitted to the conference mentioned future work that would significantly enhance the utility of the method.
This work was work that was always planned to be part of the same "project" as the work in the paper, indeed the conference paper is presented as a part way step towards doing X, and the future work is what takes it all the way.
The extension was quite involved (and not quite complete when I submitted the paper for review). They day before I left for the conference that work was "completed", in that a journal article about it was submitted for review.
Now when I present at the conference, in the conclusion I want to mention the future work – at least to the extent that it was described in the conference paper. Should I go into more detail? Should I even mention that "This work is now complete, and is under review for publication"?
The material is highly relevant to your presentation so you don't need to ask people on the internet whether it should be included.
You can and should mention it. Your talk should be based on the current best understanding of the subject, whether that comes from your conference paper or from a subsequent journal submission. It would be silly to give a talk about how, for example, your experiments show that gravity causes apples fall from trees and you're planning future experiments about its effect on other fruit, if you'd already done those experiments and, in fact, submitted a journal paper about how gravity causes all massive objects to be attracted to one another. Indeed, I've often been to conference talks where people have presented stronger results than they'd originally submitted. That's just a natural consequence of the submission deadline being some time in advance of the conference.
In a comment, you mentioned that you were worried because the originally submitted material has been peer-reviewed but the more recent material hasn't. That's a legitimate concern but not, I think, a serious one. Conference reviewing is very much "PeerReview lite™" and reviews are more at the level of "There's nothing obviously wrong with this and I think it's interesting enough to be in the conference." Journal reviewing aims more towards "As far as I can see, this is correct." Since your original submission was of an appropriate standard, it's very likely that the additional material is of a similar and acceptable standard. Given that you were confident enough to submit that material to a journal, you should be confident enough in its correctness to tell people at the conference about it.