I had the same problem. They way I solve it was a bit different, but quick.
First I created three columns next to the columns containing the cells with
the apostrophes (with numbers or text).
In the first column I inserted a number followed by an apostrophe (i.e. 0' )
in all cells.
In the secont column I merged the containt of the previous cell with the
original cell by using =(B2&B1). This created a column where all cells
started with 0' followed by the original content. I then copied and pasted
only the value of the cell. With the comand "Text to Column..." I then split
this column using ' (apostrophe) as delimiter. This inserted in the 3rd
column the original value without the apostrophe.

"Nino" wrote:

I have a large Excel spreadsheet made up of about 30,000 cells.

Most of them have a leading apostrophe (') in them, whether they are numbers
or text.

I would like to remove all the leading apostrophes without having to do it
manually, cell by cell.

Any ideas?