Is it obligatory to repeat all of the day's prayers if a piece of nail fell off during ablution and I doubted when it was on my finger?

The validity of ablution depends on removing whatever prevents water from reaching the skin. However, if the questioner does not know when this obstruction occurred, it is probable that it happened after the last prayer, and this is the most likely and probable scenario. Therefore, there is no requirement to make up the prayers, because the default for every incident is to estimate it at the nearest time. Consequently, your prayers are valid.