Finding Found Familiars
Suspicious players could use the spell Detect Evil and Good - a 1st level divination - to at least confirm there is something unusual about an animal, as a familiar summoned by Find Familiar:
is a celestial, fey, or fiend (your choice) instead of a beast.
And Detect Evil and Good tells you:
if there is an aberration, celestial, elemental, fey, fiend, or undead within 30 feet of you, as well as where the creature is located.
A rat which is actually a celestial/fey/fiend is almost certainly a familiar and at least suspicious.
Unfortunately in most respects familiar remain relatively unintelligent beasts without natural capacity for language; none of the default familiar options, including rats, know any languages and the Find Familiar spell doesn't state a general exception to that. The familiar is somehow able to understand, obey and even telepathically commune with its master, but has no means of conversing with anyone else.
The 1st-level spell Speak with Animals allows a character to communicate effectively with beasts. A familiar is not a beast, but a sympathetic GM might rule that a familiar is close enough that the spell works on them. However, a familiar who understands it is opposed to the players would probably not be cooperative, and it's difficult to threaten them since if they "die" the spirit simply disappears back from whence it came and can be summoned by its master again later.
The 2nd-level spell Detect Thoughts is normally a quite useful interrogation technique, but unfortunately it is ineffective when used on creatures with an Intelligence of 3 or less, or that do not understand language - both rule out the rat familiar.
The 3rd-level spell Tongues allows you to give a creature automatic understanding of all language and the ability to be understood by any creature that already knows a language, and it has no restrictions that would prevent you using it on a familiar. It would not be a very bright conversationalist but it should be intelligible. Unfortunately you have the same problem as above that simply being able to converse with the creature does not compel it to actually tell you anything.
However, if you're able to talk to the creature, the 4th-level spell Charm Monster (not part of the basic rules, but essentially identical to Charm Person except that it works on any creature) would convince the familiar that you are its friend and encourage it to be helpful to you, at which point it should be easily persuaded to tell you about its master.
3rd and 4th level spells are probably beyond the reach of a party going through LMoP, though.