Counting and spelling up

Pyth, 19 bytes

sXzsJ+rBG1jkUTs._MJ

Try it online: Demonstration or Test Suite

Explanation

sXzsJ+rBG1jkUTs._MJ
      rBG1            the list ["abc...xyz", "ABC...XYZ"]
     +    jkUT        appends the string "0123456789"
    J                 save this list of 3 strings in J
   sJ                 join the strings in J
               ._MJ   create all prefixes of the strings in J
              s       and combine them to one list
 XzsJ         s._MJ   translate the input string, chars from sJ
                      get translated to the correct prefix,
                      chars that don't appear in sJ don't get translated
s                     join all resulting translation strings

Python 2.7, 100 98 96 bytes

a=[]
for c in raw_input():d=ord(c);a+=range(max(d&96|1,48),d)*c.isalnum()+[d]
print bytearray(a)

TeaScript, 24 bytes 26 28

TeaScript is JavaScript for golfing

xl(#(i=lN()1)h(ii(l)+1))

Pretty short

Try it online

Explanation

x.l(#            // Loops through each character of the string

     (i=l.N()1)  // Determines whether a character is alphanumeric
                 // Will return A-Z, a-z or 0-9 depending on result
                 // Assigns to variable i

     .h(         // Get characters up to...
        i.i      // where the character is in "i"
     ) + 1       // Increased by one
)