What is the effect of an involuntary belch from a worshipper during ablution, and food rising from the stomach to the throat during fasting, on the validity of their prayer and fast?

If what you feel is merely a whisper (waswas) and you are not certain that gas has been released, then your ablution (wudu') is valid. However, if you are certain that gas has been released and it is not continuous for most of the time, then it invalidates ablution. But if its release is continuous, then it falls under the ruling of سلس ( سلس البول - urinary incontinence - which implies a continuous flow). If you did not intentionally vomit, and it reached your throat, then you expelled it without swallowing it, your fast is valid. However, if you swallowed it after it reached a point from which it could have been expelled, then your fast is broken, and you must make up that day.