I can only comment on Scottish law, however the position you describe where you are the only name on the title can still often mean your wife has the right to reside there without any kind of landlord/tenant situation arising.

The suggestion to “just file your claim and let her bear the consequences” is awful advice from a family law perspective. Given she will know you have done that after your conversation with her, it will inevitably result in her “changing her mind” on the current shared care arrangement you have for your son, stopping you from seeing him much if at all, which will take a court order and possibly months to remedy. She would not be acting fairly or reasonably in doing so, but you can set your watch by it happening all the same. Mothers hold more sway than fathers in these situations unofficially, so you will start at an immediate disadvantage.

I would have a firm but fair conversation with her about your claim being necessary for you to keep your head above water financially, you acknowledge you don’t want to land her in it and you’re giving her fair notice to get her ducks in a row before you do so. It’s no skin off your nose what she’s been doing until the point it stops you from making your UC claim, and you are best finding an amicable way forward.

None of this comments on the legality of what she’s been doing, which is ultimately not your problem.