[RPG] How to an NPC communicate with the party over long distances?

Your familiar understands you perfectly.

You've already quoted nearly all relevant text in your answer. Telepathy (within range) communicates without necessarily using words, but the target (with int >= 1) understands perfectly.

I think your confusion comes from when you state

simple animal familiar gained

There is simply no such thing. From the spell description:

You gain the service of a familiar, a spirit that takes an animal form

and thus it's impossible for it to actually be a simple animal.

Long-distance: Sending

3rd-level spell: "you send a short message... to any creature with which you are familiar.... You can send the message across any distance...." (PHB p.274)

You could purchase use of the spell from a mage-for-hire; Adventurers League guidelines^* would suggest you're probably looking at 90gp per usage.

You could hire someone with the ability to cast the spell: What's a good pay rate... might have some useful advice on that matter.

You could invest in scrolls or potions of sending. (In any world that makes a modicum of sense you've got to have 5th-level wizard drop-outs churning this exact item out as a way of making a living.) The DMG has advice on fashioning magic items that may help you and your GM.

You could train up the talent yourself—befriend an NPC, bring them along to fifth or sixth level, then station them at HQ.

Sending Stones are a similar magical item

You could invest in sending stones (DMG p.199) which allow for the casting of sending once per day; the DMG would price them at 101-500 gp (uncommon item), though the Sane Magic Item Prices index would put them at 2000 gp. (With thanks to @anaximander for the pointer.)

The spell's there; the possibilities are endless.

Intra-city: Animal Messenger, or human messenger

2nd-level spell: similarly short message, range of 50 miles, a tiny beast delivers your message in your voice. There's the possibility of your messenger getting nabbed along the way, but at half the price^* you can send redundant messages and still save 10gp!

Of course, at this distance and cost it's not clear why you wouldn't use the solution known since time immemorial: street urchins. (Available at 2cp per mile, per PHB p.159 "Services", thanks to @Sebkha.)

^{* - a list of spells available through "Spellcasting Services" is provided in the front-matter of every Expeditions module. Remove curse and speak with dead are the only two third-level spells listed, at a price of 90gp each. Various second-level spells are priced at 40gp each, divination (4th-level) is listed at 210 gp.}