An easily extendable solution:

{
   for ($text) {  # Alias $_ to $text.
      /\G ``   (.*?) ``   /gcsx && do { print($1         ); redo };
      /\G \*\* (.*?) \*\* /gcsx && do { print("<b>$1</b>"); redo };

      /\G
         (
            .            # Catchall.

            (?:          # These four lines are optional.
               (?!``)    # They are here to speed things up
               (?!\*\*)  # by avoiding calling print for
            .)*          # single characters.
         )
      /gcsx && do { print($1); redo };
   }
}
[download]

Handles mismatched ** and `` by treating them as normal characters.

Update:

• Removed "\n"s from prints.
• Added /s option to regexps, since I'm guessing newlines are not special.
• The 1st .*? was ((?:(?!``).)*
• The 2nd .*? was ((?:(?!\*\*).)*

Tested:

foreach (
   '**bold**',
   '``**bold**``',
   '**``bold``**',
   '**bold** test ** test',
   '``text`` test `` test',
) {
   print('in:  ', $_, $/, 'out: ');
   TOKENIZER: { ... }
   print($/, $/);
}

__END__
in:  **bold**
out: <b>bold</b>

in:  ``**bold**``
out: **bold**

in:  **``bold``**
out: <b>``bold``</b>

in:  **bold** test ** test
out: <b>bold</b> test ** test

in:  ``text`` test `` test
out: text test `` test
[download]