Awakened trees can speak, but are rarely described as doing so of their own volition.

This is something that's not heavily covered in canon.

In the few references I can find relating to the disposition of awakened trees, they practically always serve as guardians of individual druids, and usually have no reason to speak. The awaken spell is clear that they possess the ability to speak, and will therefore do so in order if necessary to serve their druid master.

An awakened tree appears in the D&D 5e adventure module Rise of Tiamat (p. 50), where it is the servant of a druid. It has no speaking lines.

Another reference to awakened trees appears in the Princes of the Apocalypse errata document, in a paragraph added to p. 144 of the adventure. This specific type of awakened tree is grown from a magical seed, and is described as understanding a language, but whether or not they use language is not mentioned, suggesting that it's not considered relevant:

The awakened tree understands one language known to its creator and follows its creator's commands to the best of its ability.

In earlier editions, the D&D 3e Masters of the Wild, p.37, describes awakened plants and animals as ready and able to answer questions at the behest of their master, although whether they start conversations is ambiguous:

Armed with intelligence and the ability to speak at least one language, the animal no longer needs training to understand the druid's wishes. Thus, the druid gains a source of advice and ready conversation in addition to a guard and a servant. Of course, as a fully sentient creature, an awakened animal develops its own desires and ambitions.

Treants on the other hand are more often described as talkative, and treated like NPCs, but awakened trees are typically treated as "muscle" for a druid character, rather than characters in their own right, and more or less exist to hit stuff.

Of course, this is just a generalization, and admittedly one based on very limited available data. Any given awakened tree may have its own personality, and some may be talkative or friendly.

Only if they have a friendly animal

This works, but only if they include an actual small beast in their conversation. None of the abilities involved indicate a language or other static kind of communication that could be attempted without an actual plant or animal to interact with.

That said, if any small animal is included in the conversation (for example the mouse from the Urchin background, a befriended wild animal, or a 2 cp chicken), it works more or less exactly as you describe. The Firbolg can talk to the chicken, who can tell the gnomes what the firbolg said. The gnomes can talk to the chicken, who can transmit messages between them. Someone affected by speak with animals can understand anything the chicken says to the gnomes.

Ruling that these abilities confer languages is not a bad idea

Animals in 5e don't have languages. In past editions, the ability of gnomes to speak with animals often implied that they did:

3.5: forest gnomes have "a simple language that enables them to communicate on a very basic level with forest animals"

AD&D: surface gnomes speak "the languages of burrowing mammals"

If you want to run a game with previous edition setting material, or if you want your world to have this sort of animal communication, it makes sense to have animals have languages and rule that these abilities work via that mechanism and thus are interintelligible. This is, of course, not part of the RAW but it also isn't contrary to the RAW and consequently you won't have any mechanical problems you didn't already have when ruling this way.