Relative to what? How about a clue as to your accuracy and stability requirements? The appropriate device could be anything from a free running oscillator to a GPSDO.
I also don't like the idea of what sounds like a transfer standard. With a synthesizer, you can drive whatever clock your driving directly from your timebase.
One of my early GPSDO contrivances would take about an hour to recover from every time I opened the outside shop door. That was eventually fixed by double insulating the OCXO. I also had plenty of fun trying to keep the 5 and 10 MHz outputs from leaking out of the box, and trashing my WWV reception.
How long? With all due respect, you're being far too vague here. Different technologies offer different short and long term accuracies.
That's always the case. More accuracy and stability always costs more, usually on a non-linear scale. At the high end, a small increase in accuracy and stability can cost outrageous amounts in construction costs. However, the worst cost of accuracy and stability is your time in calibrating the reference OCXO. I generally have to tinker with the OCXO for several days before it settles down. Note the differences between accuracy and stability. Also note that the Allen variance goes out to 10^4 seconds (about 7 days) and that the next order of magnitude in accuracy requires 10 times that or 70 days.
Who's time? There are subtle differences between UTC, astronomical, navigational, space, and political clocks. Einstein's time dilation is involved. Not the table at the top of the page: Looks like about 16 seconds difference between UTC and GPS time.
A mechanical wind up clock would work if you don't bother to specify your accuracy and stability requirements.
I can't tell what you're trying to accomplish, but if all you want is accurate time, and not accurate frequency, then all you need is an accurate clock. The 1pps from most GPS receivers will drive almost any kind of clock with good accuracy (as long as you don't have any missing pulses). If you don't mind the clock wandering around a bit during the day, a WWVH 60 KHz clock might suffice.
On the other foot, if all you want is a stable frequency, you can get close with an OCXO or possibly a TCXO. An OCXO is probably best, especially if you add extra insulation. Start at 10 MHz and divide down to 5, 2.5, and 1 MHz depending on what you need. I have such a system distributing 5 and 10 MHz to all my test equipment that will accept an external reference. On a good day in the ionosphere, I get about 0.01 ppm accuracy against WWV.
Incidentally, I have a pile of about 200 Novatel/CMC Allstar 12 GPS receiver boards if you want to play. They're old (2001), not the best, and don't have a frequency output, but are good enough for 1pps and for building ham-radio APRS trackers. The catch is that you'll have to build an interface board to make it useful: