The best known bounds seem to be due to Nicely (A new error analysis for Brun's constant, Virginia J. Sci. 52 (2001), no. 1, 45–55), who showed that Brun's constant is
$$
1.9021605823 \pm 0.0000000008$
$$ To do so he computed all prime twins up to $3\cdot 10^{15}$. Since the known upper bound for prime twins differs from the expected number by a factor of 4 (better bounds are known, but making them explicit would probably be extremely difficult), it is likely that the only possible way to improve on this would be to extend the range of computation.