So, the first consideration goes to the radio propagation aspect. Every country does things a little different with their radio clocks. And you might only have good signal a few times a day. For a nice overview, check out a solution for the German DCF77. If you're totally set on using a radio clock, time-nuts has discussed in the past ripping open a $10ish clock to get at the fun parts of a WWVB receiver, so that's one overview of the pool you're diving into.
For myself I've skipped past the terrestrial time keeping and went straight to GPS. Find a receiver that provides a pulse-per-second and attach that to an interrupt and process it alongside the timestamp; you can see how I've gone about that in both code and final product.
I think GPS is both easier and more precise than the alternatives, assuming you're above ground and can pick up satellites. I think of RTCs as something to tell me the day and the year, while GPS can tell me the second. But you could consider ripping the antenna out of a cheap bedside radio-controlled clock and using that to set an RTC, careful to consider whether you want a cheap RTC (DS1307) that drifts a minute per day or a slightly more expensive RTC (DS3231 or ISL1220) that drifts a minute per month. (Rough numbers of course.)