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#Japt, 61 59 92 85 84 bytes

I'm offline for most of the (long) weekend, if any more issues are discovered with this, please ask a mod to delete it for me until such a time as I can fix it.

Takes input as a string for both operations and returns a string for both as well or false for invalid input.Assumes number inputs will always contain multiple digits, add 1 byte replacing UÉ with Un<space> if that's not valid. Returns false for test case GIGATESTER but, according to the rules, that should be invalid input.

V="OIREASGTBP"UÉ?¡VgXÃe"%v"²_i1'NÃe"%V"²_i1'UÃ:!Uè"%v%v|%V%V|[^{V}NU]" ©Ur"N|U" £VaX

Try it: Numbers -> Letters or Letters -> Numbers

• 2 4 bytes saved thank to obarakon, who also convinced me to take this up again after I abandoned it earlier. I wish he hadn't!
• 33 26 25(!) bytes sacrificed implementing a quick fix (i.e., yet to be fully golfed) to check input validity.

##Explanation (Yet to be updated to the latest version)

                          :Implicit input of string U.
V="..."                   :Assign the string of letters to variable V, in order.
UÉ                        :Subtract 1 from U, which will give a number (truthy) if the input is a number or NaN (falsey) if the input is a string.
?                         :If it's a number then
¡                         :    Map over the input string, replacing each character (digit) with ...
VgX                       :      the character in string V at index X, the current digit.
à                        :    End mapping.
e                         :    Recursively replace ...
"%v"²                     :      every occurrence of 2 vowels (RegEx) ...
_i1'N                     :      with the current match with an "N" inserted at index 1.
à                        :    End replacement.
e                         :    Another recursive replacement of ...
"%V"²                     :      every occurrence of 2 non-vowel characters (i.e., consonants) ...
_i1'U                     :      with the current match with a "U" inserted at index 1.
à                        :    End replacement.
:                         :Else, if it's a string then
Uè"%v%v|%V%V|[^{V}NU]"    :    Count the number of matches of 2 successive vowels OR 2 successive non-vowels OR any character not in contained in string V plus N & U.
                          :    (The longest part of this code is the fecking input validation!)
?                         :    If that count is greater than 0 then
T                         :       Return 0.
:                              Else
Ur"N|U"                   :        Replace every occurrence of "N" OR "U" in string U with nothing.
£                         :        Map over the string, replacing each character (letter) with ...
VaX                       :         the index of the current character X in string V.
                          :Implicit output of resulting string