Frank Harrell has pretty much answered this in comments. To summarize into a formal answer:
the outcome had a follow-up time 180 days after admission...I am only looking at outcomes after patients have left the hospital...(emphasis added)
If your interest is solely in outcomes after discharge from the hospital, then a natural reference for
time = 0 is the date of discharge, with covariate values taken at or just before discharge included in your model. There is no need to set
time = 0 to the date of admission.
That limits your analysis to those who actually live long enough and are well enough before 180 days to be discharged. That limitation needs to be addressed in your presentation and any application of your model--it wouldn't apply to those who are hospitalized for over 180 days.
Yes, as the length of hospital stay varies among patients there will be different follow-up times among patients relative to their discharge dates. But that's just right censoring of survival-time observations if you are interested in post-discharge survival.
Although NRI/IDI analysis is probably not a good choice, as noted in comments, the issue of choosing survival time points for evaluating models isn't specific to NRI/IDI. For example, the
calibrate() function in Frank Harrell's
rms package requires such a choice for survival models.
There's no need to restrict yourself to a single time point. Evaluating several time points might be informative in terms of the models' handling of both soon-after-discharge events and later events. If there is some generally accepted evaluation time in your field (for example, re-admission within 30 days of discharge is often of interest), then that should be included among your evaluation times.
Finally, your model comparison is between 2 nested models, one that contains an interaction between 2 predictors and one that doesn't. Thus a likelihood-ratio test between them provides a well established and sensitive comparison, with the related Adequacy Index indicating how much including the interaction improves performance.