Are you employed? I asked this right up front because this is often the main thing that keeps women in a situation they are not comfortable with. If you're not employed, you need to be.
I know it's him that's pulling back, but how long has the sex ceasefire been going on?
Just because he has sexual feelings for men is not a hall pass on a marriage, but if you have been in a sexless marriage for very long, then all this talk about monogamy is kind of a moot point if there's just no sex going on anyway. But if that is a recent development because of him developing feelings for this man, it makes it no different than when a married man married falls for another woman.
If there has been no sex going on for a long time and you are okay with that, which is not entirely clear, that's a different situation than if you are accustomed to having sex with your husband until recently but now his affections are elsewhere.
I had a close gay male friend that I've sort of been through this with after he married a woman. When I knew him he had married a first wife young and divorced, but then the whole time I knew him, he seemed exclusively gay. So I was surprised when he fell for a woman and married her, so he must have been bi. But he was attracted to more men than women.
After they had been married for some years both of them were having horrible health problems, surgeries and all, and so sex was very limited and probably just not happening a lot of the time. He started getting a yearning for men again and would tell me this over the phone because he lived in another state. And then the worst thing was he had not told his wife that he used to live a gay lifestyle, so that came out and she was like, Don't ever touch me again.
I actually went and spent some time with them and talked to them both. They had a nice life together except for all these horrible health problems which eventually killed him, a cancer undiagnosed because they attributed the pain to his hip replacement.
But for her, the sex was over entirely once she found out, and of course she was very unhappy with him for hiding that from her. I do not know if he went and cheated on her, but it wouldn't surprise me.
They did not have children.
You need to be honest with him about how this would affect you staying in the marriage and/or your sex life. You should be very blunt with him if going to someone else for outside sex or relationship is a total deal breaker that will end in divorce so that he knows the consequences.
This thing may fizzle, but I would kind of be surprised if it fizzled before something sexual happened between them, but then I don't know your husband and his ethics.
It's a complicated situation with the children and with you trying to sort out how you feel about it and what you should do and what sort of stance you should take. But you should concentrate on what YOU would want to do about it more than his problems and keep him up to date on how this is going to affect everything and what you will and won't stay put for.
Here is a link about shared child custody norms in the UK.
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Like the US it basically says it is trending toward 50/50 custody.
I would just tell you not to panic and have a knee-jerk reaction thinking that you should get sole custody or most of the custody or anything like that, and I say that for two reasons.
One is that the kids still need their father. And one is that it would leave you with an unfair burden. So don't let your fear of the situation prevent you from seeing to it that he still takes 50% responsibility for spending time with and raising the kids. The way it's typically done is the weekdays and the weekends are split equally between the parents so that not just one of the parents has to see to getting the child to school and doing all the daily practical stuff for the child but that those responsibilities are truly shared.
Another big thing you have to address with him and maybe legally is when either divorcing party with child custody has the children, it is never good to have strangers that you may or may not be dating in the house around them. It's not safe, male or female. So if it ever comes down to that, you need to get some kind of agreement that the children will not be introduced or have strangers around them until there is plenty of time to know who those people are. And that goes for you as well as him.
If he realizes the reality of the situation of sharing custody and how much that will affect his lifestyle, he may lose any desire to disrupt the marriage. Right now I bet anything his mindset is that you will just take care of the kids while he goes and does what he has to do with this relationship. But if you let him think that he can go off and do whatever he wants and then see the children when it's convenient and no responsibility getting them to school and seeing to their meals or taking them to the doctor, that is only enabling him. So you keep sharing custody on the table and be sure he realizes that he will be responsible for them 50% of the time or close to it and that you will not be an on call babysitter but will stick to your custody schedule.
If I were you I would even show him that article so he starts realizing what reality is like and then I would talk to him about no strangers around the kids.
At least if you did a formal divorce you could get all these agreements legalized and in writing so that you wouldn't be haggling about it like you would if you just decide to open the marriage because then there is no provision for child custody and who takes care of the kids.
The 50/50 custody arrangement gives you both an opportunity to work and support yourselves. And you're going to need to work if you aren't already.