I'm asking specifically about Hilton properties, but you can answer generally too. Once I check in to a hotel room online or in the hotel app, can I still cancel my reservation, for instance, if my flight is delayed as long as I'm within the hotel's cancellation window (often 6PM same day)?
Firstly, note that most Hilton properties now have a "day before arrival" cancellation policy on most reservations. Some corporate rates override this and still have a 4pm/6pm/etc policy.
However if you are still before the cancellation time on your reservation then you can still cancel even if you have checked in online - however you will need to contact the hotel directly to do so.
The Digital Check-in FAQ covers this to some extent where it says "Should you need to cancel a reservation, please call the hotel directly. We plan to enable in-app cancellation in the near future."
To answer generally, there is the official policy, and then there is what the hotel chain or individual hotels do in practice. If it is before the cancellation deadline, then officially you are permitted to cancel. If it is past the cancellation deadline, then you need to contact the hotel chain guest assistance desk or the individual hotel directly, and plead your case. If you have a pre-paid, non-refundable rate, it's more difficult, but sometimes an exception is made. If you need to cancel because your flight is delayed or a similar circumstance out of your control, you may be permitted to do so. If you ask to have your reservation shifted to be a day later instead of cancelled, that's often easier for a hotel to do. In my own experience, I've always been permitted to shift or cancel when a flight delay or bad weather caused the issue.