Hi, yes it does sound complicated and I'm not sure I've fully understood all your questions, so I think you really need to discuss it with a mortgage broker/your bank/a solicitor to get it sorted!
That being said, you do realise that because you are married, your husband already effectively owns half of your assets including the house as a joint asset of the marriage, regardless of whether he's named on the mortgage or whether he's 'paid you back' or not? The mortgage is just a debt secured against the house, it doesn't determine what would happen in the event of you splitting. I'm not sure what the 'agreement' you have that you get the house if you split is but pre-nups even if properly drawn up and legally notarised are not legally binding in the UK, and if it's a verbal agreement it means diddly squat in the event of a contested/acrimonious divorce I'm afraid. So your plan to 'reward' your DH for sorting himself out financially by 'allowing' him to take on half the debt seems a bit wonky to me (no judgement here, obviously you chose to marry him so that's fair enough, but it's just how marriage works - for richer for poorer yes, but also 'with all my worldly goods I thee endow' - not 'with the part of my worldly good I trust you/you've 'earned' I thee endow'!).
I assume that because your Mum is liable for part of the mortgage, she also therefore is named as a joint tenant or tenant in common on the deeds, and therefore owns at least a part of the house? Did she contribute to the deposit/equity and is she expecting that to be paid back at any point? What was your plan to eventually buy her out or remove her from the mortgage at the end of the term - were you married when you took it out? If your main concern is to protect the house in the event of a split from your DH then by far the best thing would be to leave it in joint ownership between you and your Mum, because in theory then only the part of the house you own would be a marital asset and therefore he'd get much less (assuming the judge accepted that your mum genuinely does own part of the house, and also how assets are split in a divorce is based on need and not necessarily a straight forward 50:50, but obviously if there's less in the 'pot' to start with then he'll get less whatever the ratio of the split).
On your salary I would have thought borrowing £138,000 would be OK, although you don't mention how much the house is worth, the absolute maximum loan to value rate for most providers is 90%, although you'll get a better deal the lower the LTV. They will probably enquire about your marital status and your husband's debts (which, because you are married are to an extent joint debts) and for a lot of lenders it will simply be impossible for you to get a sole mortgage when you are married, particularly for a high LTV rate (because as above you can't be viewed as the sole owner) so this is something you need to discuss with your broker.
As to how to approach it with your Mum (and assuming you don't want to tell her about the debts), I'd simply say as you have said that now you are married and with a baby on the way it's time to regularise the financial situation. That wouldn't be a lie per se in my mind, just an omission...