Following a power recycle and change of internet router, my Time Capsule will not appear in Airport Utility and flashes amber.
This is normal and expected. The Time Capsule is still programmed with the specific settings that it needs to connect to your "old" router. So, it is looking for your old router which is no longer there, cannot find it, and does not know what to do.
The solution is to perform a Hard Reset or Factory Default Reset and then set up the Time Capsule again from scratch, so that it will detect and pick up the new specific settings that it needs from your "new" router.
A Hard Reset or Factory Default Reset will not affect any of the stored data on the Time Capsule, and you can use the same device name and passwords that you used before when you set up the Time Capsule again, if you wish.