按时, 尽快, 急忙: on time, as soon as possible, in a hurry
【四07】时间副词 / adverbs
Tone colorsā tone 1á tone 2ǎ tone 3à tone 4a neutral
These three adverbs go after the subject, before the verb. 按时 says something is done at the planned time, as scheduled (按时吃药, 按时交作业). 尽快 means “as soon as possible” and looks forward, in a request or a plan (请尽快回电话). 急忙 narrates a hurried reaction that really happened (快上课了，他急忙跑进教室).
So you do not give an order with 急忙: to hurry someone, say 快一点 or 赶紧. Compare also 马上 (right away, with no idea of a schedule) and 准时 (right on time, mostly for arriving or starting: 准时到). 按时, for its part, is about a task done within its deadline.
Register: everyday, spoken and written; 尽快 is very common in work messages and service notices.
Examples
请按时吃药。
Qǐng ànshí chī yào.Please take your medicine on time.
他每天都按时上班，从来不迟到。
Tā měi tiān dōu ànshí shàngbān, cónglái bù chídào.He gets to work on time every day and is never late.
请尽快给我回电话。
Qǐng jǐnkuài gěi wǒ huí diànhuà.Please call me back as soon as you can.
我们会尽快把东西送到您家。
Wǒmen huì jǐnkuài bǎ dōngxi sòng dào nín jiā.We will deliver your things to your home as soon as possible.
快上课了，他急忙跑进教室。
Kuài shàngkè le, tā jímáng pǎo jìn jiàoshì.Class was about to start, so he hurried into the classroom.
听到孩子哭了，妈妈急忙从房间里跑了出来。
Tīngdào háizi kū le, māma jímáng cóng fángjiān lǐ pǎo le chūlái.Hearing the child cry, his mother rushed out of the room.
Common mistakes
✗ 你急忙回来！ → ✓ 你快一点回来！/ 请你尽快回来。 (急忙 narrates a real hurry, it is not an order).
✗ 他按时了 → ✓ 他按时到了 (按时 needs a verb after it).