The evil Doctor Doozit with time

Question

You awake in a small room to the sound of dozens of clocks ticking and chiming. Once you stir, you hear a familiar evil cackle over a speaker system somewhere in the room. It's the self-proclaimed "evil" Doctor Doozit, an old adversary of yours. You wonder what he has for you today.

"I see you are finally awake! Welcome to my hidden lair in the mountains. If you want to escape, you have to simply complete a couple of challenges for me. The first one is simple: tell me the time!" The doctor breaks of for a quick cackle before resuming his spiel.

"I want you to tell me the time accurate to the second. When you are ready, press the button on the wall and tell me what the time is as of pressing the button."

You look around the room. It is 4 meter x 4 meter with a small barred window on one wall. You are a bit of a camping enthusiast so you pull out your keys with a pocket knife and a compass and determine the window to be on the south wall. You look out of it and see the sun high in the sky, so roughly midday. There is a door on the north wall with the previously mentioned button on it. The east and west walls are covered in clocks with various different times set on them, obviously the doctor taunting you. Your eyes drift to a table in the middle of the room. On it is an old computer in a state of disrepair, your phone, an old chunky phone with an oversized aerial, an old alarm clock and a pad of paper with a pen. You try the old phone with no luck. As you reach for your phone, the doctor chimes in again.

"If I hear any words out of you other than the time, I will purge the room." Just as well, your phone has no signal when you switch it on anyway.

After a brief bit of work, you speak the correct time and walk through the door, much to the annoyance of the doctor.

How were you able to determine the correct time?

Answer 1

I'm guessing

the hero takes the oversized aerial from the old phone and uses it to boost the reception of his own phone, thus getting a signal and being able to determine the time either by using the internal clock or going to a site like dateandtime.com or by calling a time service (which will speak the time without the hero having to say anything). If the aerial alone doesn't work, it can be extended by using cables from the non-functioning computer.
If the old phone is not a real dinosaur, it may be a satellite phone which would greatly improve the chances of getting a signal in a hidden lair in the mountains. Satellite phones are much chunkier than regular cell phones and have large aerials.

Answer 2

• We know your phone is on and working (from to the accepted solution). A modern cell phone's internal clock will remain accurate to one second for weeks or months without synchronizing with the network, so you can just read the time off the phone (this one due to Prashant).
• Same as above, but using the older phone. As long as it was made after the mid-1990's and uses a digital protocol (GSM, CDMA, etc.), it will receive a time signal even without the ability to make or receive a call.
• Since you have a south-facing window and the Sun is visible, you can track the progress of the Sun across the sky. Follow the shadow of the window on the opposite wall of the cell, and at the moment when it stops rising, announce that it is precisely local solar noon.
• If the computer is old enough, it will have enough general-purpose discrete components for you to build a quick-and-dirty radio receiver for the 60 kHz WWVB time signal. (Just remember that two marks occur at the beginning of the minute, and the current UTC time follows in binary-coded decimal.)