Does it take more cognitive effort to tell the time when reading a digital clock instead of analog? Following a question on UX.Stackexchange about whether or not to use an analog or digital clock on a website from a user-interface perspective it occurs to me that there is probably a psychological difference between using one or the other.

My hypothesis is that it takes longer to know the time when reading digital instead of analog because as children we are taught to understand time via analog clocks (5 past, half past, quarter to etc) so when confronted with a digital clock we have to do some internal calculations to convert from digital numbers into a mental analog clock before we actually understand what the time is. ("07:47, ok so if 07:45 is quarter-to that means it's now half-way between quarter-to and ten-to eight...").

However I'm aware this may just be my own internal process I'm citing rather than being how people in general use clocks so I'm interested to know if my hypothesis is actually supported / disproved.
The answer depends of course on the factors mentioned in the comments; however, one important factor to consider is the intended purpose of reading the clock. 

An analog clock presents information in graphical form, in a similar way to how a pie chart might present information. This is opposed to a digital clock, which presents symbolic information, in a similar way to what might be found in a table of values.  

Cognitive Fit Theory
According to research by Iris Vessey (cognitive fit theory,) the least cognitive effort will be required when the presentation format of the data matches the intended cognitive use. Vessey's research shows that symoblic information, such as that presented by a digital clock, is best when one needs to evaluate a single number (for example, to answer the question "What time is it now?").  On the other hand, graphical information, such as that presented by the analog clock, is best when one desires to make a comparison between different numbers (for example, to answer the question "How much time has elapsed?").

Therefore, your example in the question (how much time has elapsed past the hour) is best answered with an analog clock, whereas if I want to know what time my alarm is set for, the digital clock on my cell phone is a better format.

I might argue that the analog clock format does not do a good job presenting the graphical information, but that is best saved for another paper.