I would try to keep things as factual as possible. I think there is a huge danger here of either the blind leading the blind, or Chinese whispers blowing things out of proportion.
It is entirely possible that the solicitor dealing with the estate has mentioned in a general way that maybe a post nup would be a good idea. Or talked to the MIL about inheritance tax planning in general. That would be a sensible conversation for a solicitor to have, and of course may lead to increased fees for them.
And then MIL has mentioned it and DH has mentioned to you.
But what seems to be lacking is a coherent explanation of why it may be a good thing for any of the parties involved. And I think, if possible, it would be good to put aside the emotion and understand the pros and cons of anything that is being suggested. And asking those questions would make it clearer where these suggestions are coming from and why.
Don't forget that MIL has presumably recently lost her mother. Either with or without her blessing, a portion of her mother's estate has come to her son rather than to her. There may be some emotions on her side tied up with this that have nothing to do with how she thinks about you.
Taking the house/mortgage/tenants in common bit first. This would not seem to have anything to do with divorce protection. IANAL but the house is clearly a marital asset, and how it is owned doesn't change that. So a change to tenants in common would be for one or both of two main reasons - to split the ownership unevenly (but that wouldn't matter on divorce, only on death) and then to allow transfer on death to pass under a will. So if the house is owned TIC but DH's will leaves everything to you, nothing has changed.
If the idea is that DH leaves it to someone other than you, then TIC would allow this to happen, but the most common thing to do in this case would be to have a life interest trust for the spouse, going to the children on the spouse's death. It protects the inheritance for your joint children from either the children of a second marriage after the death of the first spouse (a more common problem if the woman is the first to die) or from a 'predatory marriage' of the second spouse, as marriage invalidates a will, and so sometimes widows can be vulnerable to someone wishing to marry to obtain assets.
Assuming you have a long and happy marriage to DH, the people that benefit from this arrangement are your children, to the detriment of a hypothetical future spouse of yours. That doesn't seem a particularly bad idea.
The post nup is obviously more of an issue. I presume the answer to the question 'MIL, why do you want DH and I to have a post nup' would be 'so that you don't get any of this money he received from my mother if you divorce'.
And that might be a reasonable wish, and it may be what happens anyway even without a post nup. But it might be worth exploring what a likely post nup would look like.
Between a happily married couple in long marriage, a post nup should consider the needs of the couple first. If ring fencing this inheritance would mean that you would have an inferior house to your husband should you divorce, is this what he would want? If the assets that you both have currently without this inheritance are more than sufficient for the needs of both of you, would you want some of this inheritance that came from his grandmother? Would you be content for this to be placed into trust for your children on divorce, for example?
Anything anyone tells you, do your own research and don't assume it is true/correct in your situation.