Source Material

The PHB says "Elvish" on p123. There isn't any information in the Sword Coast Adventurer's Guide, where the other styles are specifically named, though "Elvish" clearly maps to "Espruar", and "Dwarvish" to "Dethek".

The Forgotten Realms Campaign Setting, a 3E book, specifically refers to Espruar, which is the proper name of the script used by Dambrathan and Elvish, among other languages. Earlier works (like the 1E FRCS) indicate Espruar was specifically/originally used by silver elves.

Other Thoughts

In the language section of the PHB, they show Elvish, Dwarvish, and Draconic character equivalents to English letters. This implies that there are letter commonalities across all the languages, even if the specific words and letter frequencies vary between languages.

Considering where most people who speak Undercommon live, it's likely that some Undercommon is also written with Dethek lettering.

The Dwarvish language of Faerun uses a runic alphabet called Dethek, whose characters are easy to etch into stone and metal, as evidenced by the runestones and way-markers found in ancient dwarven tunnels and mines. - SCAG, p103.

5e rules that speakers of different dialects of a single language can understand each other, but as a DM you're free to explore the subject in more detail

5e briefly touches on the subject of the mutual intelligibility of languages/dialects. On page 123 of the PHB:

Some of these languages are actually families of languages with many dialects. For example, the Primordial language includes the Auran, Aquan, Ignan, and Terran dialects, one for each of the four elemental planes. Creatures that speak different dialects of the same language can communicate with one another.

However, no other provisions are made for speakers of different languages being able to understand each other. This is problem that the game largely gets around by simply declaring that everything under the sun can speak "Common" and so pretty much everyone can always communicate with each other if they want to.

If you want to start ruling about whether or not different languages have any mutual intelligibility, a good first pass would be to say that languages that share the same script presumably have some shared history of linguistic development and so speakers of such related languages can understand each other somewhat. It is your game, after all, and you can decide that's how the languages work in your world. It's important to note this sort of relationship isn't always guaranteed - in the real world example of Chinese and Japanese, for instance, though the Japanese writing system is based heavily on the Chinese one, the grammar and the meaning of words are often very different, so reading between languages can be quite confusing. There's also an extremely low degree of mutual intelligibility between the spoken forms of the languages - by and large you might understand only the occasional loanword.

If you want to be any more detailed than that, you'd have to start considering the history of the world and the development of the various cultures to figure out which languages may have influenced each other or have shared origins in a root language. In your own world, of course, those details are entirely up to you. If you're playing in a published setting, there may be lore to draw upon which clarifies the matter.