Dad's medical appointment is really getting delayed.
Dad's appointment will be very late because the clinic's schedule is backed up.
The clinic is running pretty far behind schedule, so Dad's appointment will be delayed.
予約 = appointment
予約時間 = appointment time, the time of the appointment
「backed up」は色々な意味ありますが、この場合では「渋滞している」という意味です。
Traffic is backed up on the expressway.
高速道路では渋滞している。
スケジュールより遅れている場合は「running behind schedule」と言って、すごく遅れているなら「running far behind schedule」です。
My father's appointment has been pushed back quite a bit.
My father's medical appointment has been really delayed.
この場合、次のような言い方ができますよ。
ーMy father's appointment has been pushed back quite a bit.
「父の予約がかなり遅れている」
to be pushed back 「遅れる」
ーMy father's medical appointment has been really delayed.
「父の病院の予約がすごく遅れている」
to be delayed「遅れる」
ご参考まで!