Farty & Rhonda -
Is there any concern of self-harm or violence against others? No.
Does he have a medical condition for which he needs regular medication? No
Does he have any other vulnerabilities? No
Where was he last? The pub
How is their relationship? Somewhat strained from the posts of the OP.
Has this happened before? I.e. not coming back from the pub? Yes
What does the partner think has happened?
That he has gone on a bender and is too hungover / inconsiderate / cowardly to face the consequences.
She has checked hospitals - he is apparently not there.
I do not think that the OP needs to worry for his safety at this time. Clearly things can change, but the most likely scenario is that he has been persuaded by someone to have a hair of the dog and face the music later in the day...
The fact of the matter the police at this stage would do no more than check the cells / call his parents / mates / call him / recheck hospitals.
We ping mobile phones etc / go further when there is a risk of serious harm. However unpleasant the situation the OP is in, that is not the case here. She has also never suggested it is. Several posters above suggested that she should call Police ostensibly to make him feel guilty when he eventually turned up - this is irresponsible and a waste of public resources. I do not have an issue with police officers doing paperwork, but I do when it is for the purpose of point-scoring within relationships.
OP - I'm sure he will turn up sooner or later. I would go and do something nice for yourself.