I believe you are correct in stating that if a mortgage company holds an interest in a property, they are the legal owners of the property. Although this may be part ownership, due to the terms and condition of the mortgage, they have a lot of say in regard to many aspects.
For a period of time, I had some dealings with mortgage/insurance related issues and I came across 3rd parties e.g. neighbours tipping off the company, by phoning or writing with allegations. The mortgage companies legal department did enter into correspondence with the mortgage holder and I believe that action was taken.
So in your position I would certainly write to your neighbours mortgage company, advising them of the situation. They may or may not write back to you, as the letter will be viewed as purely an allegation. But it should prompt them to review the mortgage and if the mortgage holder has not advised them of the letting or appears to be in breach of any t&c's, they will write to them. The ultimate sanction as you say, is for them to ask for the loan back.
Nothing ventured, nothing gained.
Huckster