ClearAll[Unprimeable]
Unprimeable[in_Integer] := Module[{id, new, pos},
  id = IntegerDigits[in];
  pos = Catenate@Table[
     Table[
      new = id;
      new[[d]] = n;
      new
      ,
      {n, 0, 9}
      ]
     ,
     {d, Length[id]}
     ];
  pos //= Map[FromDigits];
  NoneTrue[pos, PrimeQ]
  ]
res = {};
PrintTemporary[Dynamic[{Length[res], i}]];
i = 0;
While[Length[res] < 600,
  If[Unprimeable[i],
   AppendTo[res, i]
   ];
  i++
  ];

PrintTemporary[Dynamic[{lastdig, i}]];
out = Table[
   i = lastdig;
   While[! Unprimeable[i],
    i += 10
    ];
   i
   ,
   {lastdig, 0, 9}
   ];

res[[;; 35]]
res[[600]]

lastdigit = IntegerDigits /* Last;
Print["Least unprimeable number ending in ", lastdigit[#], ": ", #] & /@ SortBy[out, lastdigit];