Is there (or can there be) a device that could measure the speed and acceleration of time?
The "speed of time" can only be measured by comparing two clocks. That needs Relativity (as noted in other answer). Light speed is fixed so the speed at which an oscillator (pendulum) relative to a differently accelerated observer gives us the "speed of time" for each. So their is no right or preferred speed of time.
Many devices require we measure the slowing of time due various aspects of relativity, this would include GPS. Certainly measuring atomic particles that would otherwise disintegrate unless time speed was slower for them makes the LHC and any accelerator a device that can measure and must take into account time dilation. Time is a really deep concept with much accessible reading. Scientific American February 2012 "A matter of Time" is a "collectors" special edition magazine collecting some great articles about this stuff. Assume it is still easily available.
In order to create such a clock a stationary point in the universe would have to be located, or at least we would need to know how we are (it is) moving in the universe in order to create it. Since we cannot find such a point or know such a thing, no such device can be created and we will have to make do with our current various chronometers.