What is the ruling on the prayer of one who intended to perform it a minute before the expiration of its time, and completed the prayer after the entry of the time for the next prayer? And is he excused for his unawareness of the time while in the bathroom?

There is no sin upon you for delaying the prayer due to this excuse. Your prayer is considered ada’an (performed on time) if you catch one rak'ah (unit of prayer) within the prescribed time, based on the hadith: "Whoever catches one rak'ah of the prayer has caught the prayer." However, if you catch less than one rak'ah, scholars have differed as to whether it is considered ada’an or qada’an (performed as a make-up). Nevertheless, it is valid to hasten to begin the prayer with the intention of ada’an when a part of the time remains.