Beat me to it!

IntegerDigits will break up the numbers, into sub-lists

IntegerDigits@{12, 35}

{{1, 2}, {3, 5}}

Flatten will break up the sub-lists

Flatten@IntegerDigits@{12, 35}

{1, 2, 3, 5}

FromDigits will make a new integer from the flattened list

FromDigits@Flatten@IntegerDigits@{12, 35}

1235

RightComposition allows the operations to be read from left to right

{12, 35} // RightComposition[IntegerDigits, Flatten, FromDigits]

1235

Another way of saying RightComposition is /*

RightComposition[
 IntegerDigits,
 Flatten,
 FromDigits]

IntegerDigits /* Flatten /* FromDigits

Make a compound operation called joinDigits, or as you like

joinDigits = IntegerDigits /* Flatten /* FromDigits;

Call the new operation on your list

joinDigits[{12, 35}]

1235

joinDigits[{12, 35, 69}]

123569