Wolfram Language (Mathematica), 141135 bytes
Min@(s=Select)[(x@#~f~60-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2x=BlockMap[f=FromDigits,#,2]&
-6 bytes from @att
Wolfram Language (Mathematica), 141 bytes
Min@(s=Select)[(x@#~f~60-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 135 bytes
Min@(s=Select)[(x@#~f~60-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x=BlockMap[f=FromDigits,#,2]&
-6 bytes from @att
Wolfram Language (Mathematica), 142141 bytes
Min@(s=Select)[(f[x@#,60]x@#~f~60-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 142 bytes
Min@(s=Select)[(f[x@#,60]-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 141 bytes
Min@(s=Select)[(x@#~f~60-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 159142 bytes
Min@(s=Select)[(f[x@#,m=MixedRadix@{24,60}]60]-a~f~ma~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 159 bytes
Min@(s=Select)[(f[x@#,m=MixedRadix@{24,60}]-a~f~m)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2
Wolfram Language (Mathematica), 142 bytes
Min@(s=Select)[(f[x@#,60]-a~f~60)&/@s[Permutations[p=#],(w=x@#;24>w[[1]]>=#&@@(a=x@p)&&w[[2]]<60)&],#>0&]&
x@n_:=(f=FromDigits)/@n~Partition~2