Having read a question about the reliability of the PHP
time function, I wondered if there was some kind of 'reliable' web API to retrieve time from.
Does it have to be a web API? Typically you'd use NTP instead. It's a well-established protocol with plenty of freely accessible servers. It's usually used for synchronization rather than obtaining the time once though. But if you use something similar to keep your system clock accurate, then accessing the time from something like PHP should be okay.
Could you explain more about exactly what problem you're trying to solve?
You may take a look at the NIST Internet Time Service and the NTP Pool Project.