Script refers to the characters used. To an Orc, text written in Dwarvish would have familiar letters, but otherwise would make no sense to him.
Your comparison between English and French is spot-on: both have the same script, but being able to read English does not help you understand French. Compare this to for example English and Chinese: not only do you not understand, even the very symbols used to write the text with are alien to you.
Rules-as-written, yes.
Your game statistics are replaced by the statistics of the chosen creature, though you retain your alignment and Intelligence, Wisdom, and Charisma scores.
Is language part of the creature's "statistics"? Yes. Monster Manual p.6 describes "statistics" thus:
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
That heading is followed by sub-headings for Size, Type, Alignment, Armor Class, Hit Points, Speed, Ability Scores, Saving Throws, Skills, Vulnerabilities, Resistances and Immunities, Senses, Languages, Challenge, Special Traits, Actions, Reactions, Limited Usage, and Equipment. So yes, language is specifically part of its statistics, and, rules-as-written, you gain it when you shapechange.
Of course, the DM is free to rule otherwise, and many DMs will rule that you don't gain languages since it doesn't make a lot of sense. But the question does ask what the rules-as-written answer, and technically, you gain langauges.
You also retain your existing languages:
You retain the benefit of any features from your class, race, or other source and can use them, provided that your new form is physically capable of doing so.
As a side-note, nowhere in the rules specifically refers to a language as a proficiency. For example, on p.14:
A background gives your character a background feature (a general benefit) and proficiency in two skills, and it might also give you additional languages or proficiency with certain kinds of tools.
Best Answer
Use an intelligence check
The character has to hear mangled or mispronounced sounds and sort out what they are in their native tongue then translate that to the common tongue. This seems like a tall order. A professional translator or diplomat might struggle with this task. Set the DC for getting the correct message accordingly.
Tiered Failure
Failing to get the whole script spot on might still yield useful information. Some adventure modules provide different levels of information for different DC rolls. E.g. A 22 is required to get the whole message spot on, but a 10 might yield the just the nouns of the message.
Teamwork
This task could utilize a variety of skills. For example, A player could argue that knowledge history check could assist in helping the reader correctly pronounce the syllables based on some know history of their shared script or etymologies.