Is it possible to modify radio-controlled clocks (so-called "atomic" clocks, like https://www.amazon.co.uk/Wall-Clocks-Radio-Controlled/s?ie=UTF8&page=1&rh=n%3A376322011%2Cp_n_feature_keywords_browse-bin%3A1249834031) to synchronise to an arbitrary (local) time signal rather than the standard time signal (i.e. MSF in the UK, WWVB in US, or any number of other time signal stations across the world)?
As I see it, this would involve two parts:
a.) Adjusting the radio receiver in the unit to listen on a different radio frequency of a locally-provided time signal rather than the default frequency
b.) Broadcast the desired time signal in the expected format on that frequency.
I've had a look at the signal standard, and I think I'd be able to broadcast a signal as described here http://www.npl.co.uk/upload/pdf/MSF_Time_Date_Code.pdf using a handful of components and an Arduino. However, I'm not really sure how I'd approach adjusting the receiver as described in a.) - or whether there's a better approach I could take? (or other flaws in my plan I haven't considered), and my Google-fu has not brought up any useful resources to suggest others having achieved this in the past. I wondered whether I'd even need to change the frequency, or whether producing a local signal on the same frequency would have sufficient power to "override" the official time signal? Any advice would be gratefully received!
To further explain my purpose; I'm not trying to improve the accuracy - I'm trying to broadcast and display an arbitrary local "stopwatch" time, unrelated to whatever the actual current time might be. (From asker's asker's clarifying comment)