ReplacePart[list0, {_, _, 2} :> Last[list2 = RotateLeft[list2]] ]
{{{12, 270}, {34, 271}}, {{56, 272}, {78, 273}}}
Alternatively,
k = 1; ReplacePart[list0, {_, _, 2} :> list2[[k++]]]
{{{12, 270}, {34, 271}}, {{56, 272}, {78, 273}}}
Using Part assignment:
list0 = {{{12, 34}, {34, 56}}, {{56, 78}, {78, 91}}};
list1 = {{270, 271}, {272, 273}};
list0[[All, All, 2]] = list1;
list0
{{{12, 270}, {34, 271}}, {{56, 272}, {78, 273}}}
A few additional alternatives:
MapAt[Last[list2 = RotateLeft[list2]] &, {All, All, 2}] @ list0
k = 1; MapAt[list2[[k++]] &, {All, All, 2}] @ list0
SubsetMap[list2 &, {All, All, 2}] @ list0