The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.


But there are known explicit bounds for the largest prime divisor. The best one seems to be $(1+3^{d/2})^2$, where $d$ is the degree of the number field, due de Oesterlé (1994!). But as far as I known, the proof of such a bound remains unpublished.


Quoting a relatively recent survey on the topic for context ("Torsion subgroups of elliptic curves over number fields" by Andrew Sutherland, 2012):



> 
> *Oesterlé's bound plays a critical role in several of the results
>  discussed here; it is quite unfortunate that no proof has been
>  published. The work of Parent in implies that Oesterlé's bound holds
>  for all suffciently large d, but we are typically interested in
>  particular small values of d (e.g. d = 5; 6; 7). **There is current work
>  in progress aimed at addressing this gap in the literature** [6]*.
> 
> 
> *[6] Maarten Derickx, e-mail regarding innitely many rational points
>  on a modular curve of degree Q-gonality 2, December 2012.*
> 
> 
> 


* My question is, has this gap been filled? Alternatively, has Oesterlé's work been superseded?



Oesterlé's unpublished bound on Uniform Boundedness
Yes, this is published as appendix A to chapter 3 in Derickx' PhD thesis available here: <https://openaccess.leidenuniv.nl/handle/1887/43186> . The thesis contains, of course, many more interesting results.


