I agree that the sentence is a bit problematic. If we agree that
A is second only to B.
states that there is no other thing to which A is inferior in ranking, only B, which in turn logically strictly implies that B is ranked first and A is ranked second. Also, to clearly state - it is talking about absolute ranking within the scope of comparison. (Original quote might not be comparing all the countries in the world, but from the pool of compared countries, the rankings which are implied are absolute; due to the word "only").
So, if1 we agree on the above logic and meaning, let us look at
A is second only to B and C.
Here we have a problem - if we decompose to “A is second only to B” and “A is second only to C” then we run into contradiction, according to the logic I presented this would imply that A is second and B and C are both ranked first.
However, if we skip this strict logic for a second, the only other interpretation that is possible is that A comes right after B and C, which are ranked higher than A2. In this case however, as OP suggest, strictly logically we might be tempted to denote the absolute rank of A, which is indeed third, but that kind of notation is not so common (ngrams, keep in mind that expression “third only to” is represented only in portion of the results).
Still, though it is not as common as “second only to” there are quite a few instances, such as this:
third only to the Soviet Union and South Africa in 1985 in terms of output
This denotes absolute third place, stated exactly as OP intended, and after reading it a few times I come to opinion that it is quite natural and that the original should have been written like that:
India's army, in numbers, is third only to the China's and America's.
1 If we don't agree see the usage in books.
2 There would be no problem if the original quote was “India's army is second to America's and China's”, (second meaning inferior in rank) the problem occurs with introduction of the word “only”. As stated before “second only to”, as phrase, denotes absolute position; in the meaning of inferior in ranking there would be no problems to use it: “India's army is inferior in ranking only to America's and China's” is semantically perfectly fine.