Stepping Through Time
JavaScript (ES7),  136  132 bytes
Takes a single integer as input and returns a string.
s=&gt;(V=g=(d,t=s,k=0)=&gt;k&gt;V?R:[...S=([1e3]+t).slice(-4)].some(p=c=&gt;(p-(p=c))**2-1)?g(d,t=(t-~d)%2400,k+=t%100&lt;60):(V=k,R=S))()&amp;&amp;g(2398)

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How?
The only valid hour prefixes are 01, 10, 12, 21 and 23. It means that any invalid minute count starting with a digit greater than 5 will be rejected by the test on the absolute difference between two consecutive digits.
So there's no need to explicitly handle the wrapping of minutes modulo 60. Instead, we can just make sure that the distance k is not incremented on invalid minutes:
k += t % 100 &lt; 60


JavaScript (Node.js),  148  146 bytes
Takes a single integer as input and returns a string.
This version picks the answer from a hardcoded table.
t=&gt;0x60553BCD48A3CE4F56D818FDC473FC726398E907D3A9n.toString(6).substr([22,85,9,423,462,x=98,x,9,9,413,461,9,x,x,9,9].findIndex(d=&gt;(t-=d+2)&lt;0)*4,4)

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How?
Once converted to base-6, the integer 0x60553BCD48A3CE4F56D818FDC473FC726398E907D3A9n is turned into the following string, which holds all possible valid answers:
&quot;23450101012101231010101212101212123212342101212121232321232323432345&quot;

The correct answer is selected according to the interval in which the input time falls:
 answer | input interval
--------+----------------
 &quot;2345&quot; | 00:00 - 00:23
 &quot;0101&quot; | 00:24 - 01:10
 &quot;0121&quot; | 01:11 - 01:21
 &quot;0123&quot; | 01:22 - 05:46
 &quot;1010&quot; | 05:47 - 10:10
 &quot;1012&quot; | 10:11 - 11:10
 &quot;1210&quot; | 11:11 - 12:10
 &quot;1212&quot; | 12:11 - 12:21
 &quot;1232&quot; | 12:22 - 12:32
 &quot;1234&quot; | 12:33 - 16:47
 &quot;2101&quot; | 16:48 - 21:10
 &quot;2121&quot; | 21:11 - 21:21
 &quot;2123&quot; | 21:22 - 22:21
 &quot;2321&quot; | 22:22 - 23:21
 &quot;2323&quot; | 23:22 - 23:32
 &quot;2343&quot; | 23:33 - 23:43
 &quot;2345&quot; | 23:44 - 23:59

The lengths of the intervals in minutes (minus 2) are stored in an array.