There's nothing about this that someone in Babbage's time could not have build. Removing the negatives from that statement ... this could easily have been built in Babbage's time.
However, it would not have been. A literal Turing machine needs a huge tape and a very, very long time to be able to do useful calculations. Babbage's Difference Engine could be seen to be immediately useful performing a task that was essential at the time. Babbage's proposed Analytical Engine was less obviously immediately useful, but it could do things that were obviously on the way to be useful, so might have got some speculative funding.
But a naked, vanilla Turing Machine is so obviously not (directly) useful, and no one of the time would having funding building one that was big enough to do anything of practical interest.