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 https://link.springer.com/content/pdf/10.1186/s40687-014-0012-7.pdf 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).

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

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).