It really depends on the model of pickup, as well as the winder. Since the people running that company have marginalized Seymour, I've not been a fan of that company for years. However, I would buy pickups from MJ, even if it meant paying more for her to wind a production model...and yes, they sounded better. A big difference is that they put production models on a machine that is programmed with the specs and a machine does the winding. You can lose the character that way. I have some production models that she used a scatter wind for me and they are incredible.
But....
A Suhr SSH+ is an outstanding JB-killer. So is the DMT Baker Act.
For BKP, you might get some people thay suggest the HD to replace a JB. I've never heard it, as it is my opinion the highs are a bit softer...so it might fit if you are looking for an earlier version of the JB from maybe the late 70s or something like the JB Prototype or The JB Model. I find it also a little closer to the DiMarzio AT-1.
For my ears, I find the Rebel Yell to be the closest thing BKP has to offer to a JB. Not saying they are trying to copy any thing, but just how it sounds to me. Especially considering SS used a JB for so long, that maybe his hears are tuned in to that tone as well.
There are several BKP models that could suit your needs. Do some checking around and ask plenty of questions. You can also start by installing the Aftermath and seeing where that leads you. Only you can determine if any brand or model is for you, so let your ears guide you.