When doing presentations that are joint with other institutions/individuals (or are supported by them), the other parties should be acknowledged, usually at the beginning and end of the talk. Presumably you are working on the presentation at your current institution, even if it is just fine-tuning the presentation and making other small changes. Because of this I don't think it's a problem to acknowledge your current institution and use their slide templates. But, I would recommend having the name and/or logo of all institutions that have contributed to the work in the slides.
Personally, I list collaborators and sponsors on the first slide and I acknowledge their contributions to the work being presented before I actually do the presentation. In your case I would say something like "I'm currently in position X at institution A; this is research done collaboratively with institution B along Y and Z who are at institutions A and B respectively." Putting this on the first slide allows everyone to see this before your start your talk. But, I usually try to put it again at the end as well. It is very hard to do research without other support, and so that support should be generously acknowledged.