Find found pets
Suspicious players could use the Detect Evil and Good spell – a first-level divination – to at least confirm that an animal has something unusual, such as a pet summoned by Find Familiar:
is a celestial, fey, or daemon (your choice) instead of a beast.
And Detect evil and good you say:
if there is an aberration, celestialelementary, fey, daemon, or undead within 30 feet of you, as well as the location of the creature.
A rat who is actually a celestial / fairy / demon is almost certainly a familiar and at least suspect.
Unfortunately, in most cases, pets remain relatively unintelligent animals and unable to speak; None of the usual default options, including rats, know any language and the Find familiar the spell does not state a general exception to this. The familiar is somehow able to understand, obey and even telepathically commune with his master, but has no way of conversing with anyone.
The first level spell Talking to animals allows a character to communicate effectively with the animals. A pet is not a beast, but a sympathetic GM may decide that a pet is close enough for the spell to work. However, a pet who understands that he is opposed to the players would probably not be cooperative, and it's hard to threaten them because if they "die", the mind simply disappears from there. Where he came and can be recalled by his master later.
Detecting level 2 thoughts is normally a very useful interrogation technique, but unfortunately it is ineffective when used on creatures with an Intelligence of 3 or less, or who do not understand the language – both exclude the familiar rat.
The third level tongue spell, Tongues, allows you to give a creature an automatic understanding of all languages and the ability to be understood by any creature who already knows a language. No restriction would prevent you from using it in a familiar way. He would not be a very brilliant talker, but he should be intelligible. Unfortunately, you have the same problem as before: the simple fact of being able to converse with the creature does not make it tell you anything.
However, if you are talking to the creature, the 4th level charm monster spell (not part of the basic rules, but identical to Charm Person except that it works on any creature) will convince the familiar that you are his and encourage him to help you. At that point, he should be easily persuaded to talk to you about his master.
The 3rd and 4th level spells are probably out of reach of a group that goes through LMoP.