In our diocese (quite close to Salford, hello Mark

), there is an application form but it wouldn’t disclose anything like living with a partner. It covers administrative details such as name, address, d.o.b etc then asks for two referees who could vouch for you.
There is also a CRB check (to disclose any criminal records).
The reference form asks general questions about the person’s suitablity to be an EMHC e.g. trustworthiness and other generalised good qualities, but I would think it unlikely that anyone would comment on the applicant’s living arrangements since you wouldn’t choose someone to be your referee unless you thought they would write you a good reference. I suppose it could happen though, particularly if the applicants were so ignorant of the Church’s teaching on marriage and relationships.
They then go through the course which is run by the diocese to train people (some parishes still train their own EMHCs but this is being heavily discouraged by the bishop).
If, as you say, the priest already knows about this and has still chosen to make them EMHCs, you can either:
a) Assume that you might be wrong about their lifestyle and that the priest knows more than you about their situation. There may be details that you are unaware of which substantially alter the nature of your suspicions, or
b) Complain to the bishop, if you have done your best to have the matter resolved within the parish and have failed to make any headway.
Your choice. My post was mainly aiming to show that, although there is an application process, it won’t always result in the authorities finding out information such as you have outlined. It is the duty of the parish priest to choose suitable people to fulfil the role of EMHC. Only you, who have the fullest knowlege of the circumstances, can decide whether your pastor has done that or not.