My thoughts: A Bonisagus would never, in her right mind, take an apprentice that has already been initiated into a Mystery Cult House, specifically House Bjornaer or Verditius.
Let's say, for sake of my argument, that the Bjornaer apprentice has already been initiated into the Outer Mystery of House Bjornaer by the Ritual of Twelve Years, and he can change into his Heartbeast. Verditius apprentices gain their Outer House Mystery in the same season that their arts are opened. So you have two apprentices that already know important secrets of their Houses.
Should a Bonisagus choose to take one of these apprentices for her own, I would see her becoming the target of mutliple Wizard's Wars. Both House Bjornaer and Verditius have statements in their chapters about how they will violently go after anyone taking secrets out of the House. Apprentices are not protected under The Code, and can readily be studied to discover the workings of a House Mystery. Verditius and Bjornaer couldn't take legal action against the Bonisagus because it would go against the Oath.
Should an apprentice survive to become a magus within House Bonisagus, I would think she would immediately recieve, if the Mystery Cult House is generous, a "Join or Die" from that House. Bonisagus magi are required by law to share all magical knowledge that they have. Even if the new magus didn't intend to share, a Tribunal ruling could demand the knowledge. House Bonisagus might declare that this knowledge be too dangerous to the Order to spread, but that doesn't stop it from being outside of the Mystery Cult House. The House could threaten to leave the Order should this knowledge continue to survive outside their House.
I think this could realistically spark another Crisis in the Order.
Any character, PC or NPC, that has been "derailed" from one path by being claimed mid-apprenticeship by Bonisagus can make for an interesting twist in a background. These are two of the more interesting possibilities.
I forget any broader interpretations in canon, but that's not how the Code reads. They are sworn to share "all that they find in their search for wisdom and power" - I might suggest that's only once they become Bonisagus, and any previous secrets are theirs to keep.
Yes, but one would hope that a Bonisagus is above such petty larceny, and too wise to risk such. Doesn't mean either House (or individual Parens/magus) would trust them that far.
But more to the point, such Mysteries are usually taught only towards or at the end of apprenticeship, to avoid just this situation. Besides Bonisagus, there is always the chance of an apprentice failing, or "going rogue", and taking such secrets with them - which is why Parma is usually (always?) taught only in the last season(s) before gauntlet. And few Boni would be so crass as to claim an apprentice in their 15th year - the need to do so would be rare indeed.
So, between those, the situation should rarely come up where it's is not completely unacceptable.
Its important to realize that Mysteries are not just "information". They are transformative experiences. A Verditius or ex Verditius could not teach what he knows to others without that transformative experience and the commensurate change in the student's magic.
A Bonisagus with an initiated apprentice would be constantly butting up against that changed nature of their magical Gift in their attempts to teach the Bonisagus methodology.
The other othing to consider is that while a Bonisagus magus (not apprentice) has to share the results of his studies, nothing compels anyone to be in House Bonisagus. If an initiated apprentice is taken by a Bonisagus, that apprentice could chose to join the Mystery House when it comes time to swear the oath. Most of the Mystery Cults consider anyone with knowledge of the Mystery to be part of their House regardless (definitely Criamon and Bjornaer do).
Finally, the Bonisagus magus is not exempt to enmity of the House whose secrets he is trying to pilfer. The extreme example is of the Bonisagus maga who kept taking apprentices from other magi when her last one "had a lab accident". After House Bonisagus refused to order her to desist, the maga recieved Wizards' War declarations from the former master of her current apprentice and the masters of six previous (now deceased) ones.
Though there is that case of wizard war there from bonisagus abusing. Anyone that wizard war just because a bonisagus took their apprentice and was not abusive would get some strong disapproval because giving up apprentice IS part of the code and merging mysteries into the hermetic theory is part of what the order was founded for. Verditious, Criamon and Merinita though are good about it. Heartbeasts might be more tricky, especially for apprentice whose gathering of 12 year is their 4th or 5th year of apprenticeship.
I do not think that any former mystery-apprentice taught by a Bonisagus will be able to relate the secrets to other magi. While being subject to some ritual, he cannot simply impose it on others. To successfully initiate an apprentice into Verditius Magic you need a score of 3 in the cult lore (IIRC). Once being member of another house you will not get these secret teachings. You might embark on a journey to learn it by yourself, but this will take time which might be used better otherwise.
As for sharing the knowledge, I do not think you can force a Bonisagus to share knowledge he does not understand/have without the sufficient cult lore. And even if forced, Bonisagi can claim not to have finished their researches and thus do not need to share the incomplete knowledge. Any Bonisagus might even stop pursuing an aim and does not need to publish his results generated so far. Let us say breakthrough points generated before the project is left for good.
However, even if my points are not true or do not meet your interpretation of the RAW, I think any secret of a mystery cult published by a Bonisagus will lead to two consequences:
1.) The magus will be target of a multitude of WW, which will be his end, most likely.
2.) The script will end up in the prohibited section of Durenmar's library and no one will be allowed to have a look at it.
So most Magi being an apprentice to a mystery cult before becomin Bonisagus will not investigate the matter.
However, there might be exceptions that generate story potential...
In our sagas, he assumption was that the outer mysteries are already integrated into hermetic theory, but hadn't spread throughout the order yet because of the inherent disadvantages.
They're treated like any other house virtue and may be purchased at by characters of any house at character creation.