I think the answers are
(1,4) or (3,4)
The first statement means that there are no numbers that could sum together to be Timothy's such that their product would determine their sum. If any product determines either the sum or the numbers, then this cannot be the number. This eliminates:
Any sum greater than or equal to 12. If the sum is equal to 12, then the product 11*1=11 is uniquely defined as those digits. If the sum is greater than 12 but less than 23, then taking one of the digits to be 11 and the other digit to be sum-11 causes a situation where Joseph knows the number. As you cannot multiply 11 by 2 and get a possible number (and since 11 is prime), the factoring of this number is unique given the constraints. For sum >23, do the same thing with 19 (also prim) and the factoring idea is the same. This leaves only 20 and 20, which I think is obvious that the product could only be determined by those two numbers.
For 10, consider the digits 3 and 7. The product is 21, which is uniquely determined by these digits and Joseph knows.
For 8,6,4,3,2 use the property noted by Agawa where any sum that is a prime has a possible product that is uniquely determined.
We are left with possible sums of 11, 9, 7, and 5.
For these sums, we need to consider each case. A possibility will be any sum that creates only one product where the sum of digits is a number that has not been eliminated. This leaves the choices above.
For example, if the sum were 7, then the possibilities are (1,6), (2,5) and (3,4). If the numbers were (1,6) then the product is 6. This could be obtained with either (1,6) or (2,3). The sum of (2,3) is 5. As Timothy would have claimed that Joseph couldnt' determine the numbers for either of these sums, Joseph cannot determine what the numbers are.
If the numbers were (2,5) then the product is 10. This could be obtained with either (1,10) or (2,5). Timothy would have similarly claimed that Joseph couldn't determine the numbers for either of these sums, and so Joseph cannot determine what the numbers are.
If the numbers were (3,4) then the product is 12. This could be obtained with either (1,12) [sum of 13 -> Timothy believes (11,2) is a possibility which means he would not claim Joseph couldn't determine numbers], (2,6) [which again has a sum of 8, which could be obtained by (7,1) and Timothy would not claim Joseph couldn't determine the numbers] or (3,4) [sum of 7. Timothy WOULD claim Joseph couldn't determine the numbers]. As (3,4) is the only possibility for which Timothy would claim Joseph couldn't determine the numbers out of this product, then Joseph knows this is the answer. As none of the other options for 7 lead to Joseph knowing this is the answer, now Timothy knows this is the answer too.
For 9, it breaks down because for both 1,8 and 2,7 Joseph would be able to determine the answer, which means that Timothy couldn't figure out which it was based on Joseph knowing.
For 11, it similarly breaks down for (3,8), (4,7) and (5,6).
For 5, only the choice (1,4) leads to Joseph determining the answer, so Timothy can figure this out too.
EDIT: changed first elimination to state SUM instead of numbers per Duncan's comment. Also updated first algorithm to change 20 to 19 and add note about primality of 11 and 19. Also note that it's not possible to determine which number is S and which is G as both addition and multiplication are associative. You can only figure out solutions where either number could be either number of coins.