The Clock Hands of Doom
JavaScript (Node.js), 54 47 bytes (round to the nearest)

-7 bytes. Thanks @user202729



a=&gt;b=&gt;[(a+=b&gt;5.46*a)+a/11|0,a%12*65.46%60+.5|0]


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JavaScript (Node.js), 40 33 44 bytes (rounds towards 0)

-3 bytes thanks to @Arnauld

-4 bytes thanks to @Kevin Cruijssen



a=&gt;b=&gt;[(a+=b&gt;5.46*a)+a/11|0,a%12*65.46%60|0]


Explanation

a=&gt;b=&gt;[(a+=b&gt;5.46*a)+a/11|0,a%12*65.46%60|0]    Full Code
a                                               Hours
   b                                            Minutes
    =&gt;[                    ,               ]    return array with
       (a+=        )                            add to the current hour
           b&gt;5.46*a                             1 if the minute's clock hand has 
                                                passed the hour's clock hand. Here we use
                                                equation 60*a/11 which is the same as 5.46*a
                    +a/11                       and add 1 when hour is 11
                         |0                     floor the result

                            a%12*65.46%60|0     Here we do equation ((720/11)*a) (mod 60)
                            a%12                In case of hour 12 we take 0
                                *65.46          multiply hour by 720/11 which can be shortened to
                                                65.46 to save 1 byte.
                                      %60       mod 60
                                         |0     floor the result


Side note: I'm pretty sure this can be golf down by someone with more knowledge at math. I barely know how to sum and multiply

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