As I understand matters, the only way to get an explicit bound for gaps between consecutive primes (not strings of $m$ consecutive primes for some $m\geq 3$) using that particular level of distribution that is optimal relative to the method is to completely rework everything in 

* DHJ Polymath, _Variants of the Selberg sieve, and bounded intervals containing many primes_, Research in the Mathematical Sciences volume 1, Article number: 12 (2014), doi:[10.1186/s40687-014-0012-7](https://doi.org/10.1186/s40687-014-0012-7)

with $\theta < 0.8$, computing a large number of the implied constants and choosing a new admissible set relative the the level of distribution you have selected.  Computing the implied constants is straightforward but tedious.  Finding the optimal admissible set sounds a bit more computationally expensive (but completely doable).