Rich said:
Well, DigiKey has a relay (several, actually) with a 12V, 960 ohm coil,
that'd be a time constant of 4.512 S, which would hold up for a glitch,
but that's about it.
Now, if you really insist on letting a cap discharge, just amplify
it.

Use a MOSFET, although you'd probably want to use another
transistor for a little positive feedback so you'd get a snap
action - I don't know if you'd want your MOSFET to be in the linear
region for 2-3 minutes.
Cheers!
Rich
The brief delay is all he needs. He can sense the relay with
software once every desired_delay_time and won't need a routine
that says if power bad, sense again in 4 seconds, if power
still bad, signal.
However, since people seem to think the full delay needs to
be in hardware, the cap amplifier is a good idea. I did that a
long time ago, with bjt. Here are two circuits, aimed at the OP's
self-described level of familiarity (and perhaps a bit beyond it).
I think the first is the simplest of the two for the OP to get
longer hardware delay - it yields about 1 1/2 minutes delay as
drawn.
1N400x
+----|<-----+
| |
| 5V | 166 ohm relay
+12 VDC---7805 ---[10R]---+---Relay---+
| | |
| === 1F 5.5v |
| | |
Gnd --------+-------------+-----------+
For more delay, he can parallel another supercap or super caps.
======================================================
It gets a bit more complex below, for some newbies.
Converting the relay circuit to transistor driven, he needs 2
power supplies - one to sense the line drop, the other to power
the circuit.
Here's an actual circuit (slightly modified) that is in use with a
30 amp Bosch (88 ohm AIRC) automotive relay, using a power darlington:
D1
+------------>|----------+
| D2 |
| +----->|----+
| | | D3
Power | +---[Ry1]---+---|<--- +12 from UPS
Sensing | | powered
Wall Wart | / wall wart
+12 ---+----+---[R1]---| Q1
| | e\ +-------- Gnd
[C1] [POT]* | |
| | | |
Gnd ---+----+------------+-----------+
C1 - 4,700 uF 16v (470 used in original for ~ 40 S delay)
D1-3 - 1N400x
Q1 - TIP 120
POT - 100K (* = if desired for delay adjustability)
R1 - 4.7 K
Ry1 - Any 12 volt relay.
The use of wall warts is specifically intended to avoid using
any circuitry inside UPS. From his post, it is highly doubtful
that the OP would have the confidence or capability to tap into
the existing circuits.
Ed