I'll use links to technical articles to minimize this since that's longer than you think:
Electrum will have to derive your 'Master Private Key' to start its keypool derivation.
From your 12-word 'Seed Phrase', it does these exact steps to get your 'Seed' except the default passphrase is "electrum" instead of "mnemonic".
Mnemonic to Seed: https://learnmeabitcoin.com/technical/keys/hd-wallets/mnemonic-seed/#mnemonic-to-seed

With the 'Seed', create the 'Master Private Key'.
Seed to Master Private Key: https://learnmeabitcoin.com/technical/keys/hd-wallets/extended-keys/#master-extended-key

Then with the Master Private Key (m), derive the Extended Private Key at: m/0h.
Notes: You can get it with getmasterprivate() command in the console or CLI (Electrum calls it "Master Private Key");
Its pair 'Master Public Key' can seen in "Wallet->Information" window and its derivation path.

With the Extended (Master) Private Key (m), derive the 1st to 20th receiving addresses with the derivation path: m/0/0 to m/0/19
And the 1st to 10th change address with the derivation path: m/1/0 to m/1/9 (generate more if the last one is used)
(you can check that with each address' "Right-click->Details" in 'Addresses' tab)
Child key derivation is explained here: https://learnmeabitcoin.com/technical/keys/hd-wallets/extended-keys/#child-key-derivation

But since you're going to create a tool, skip the terminologies used by Electrum and go straight from the actual master private key created from the seed.
Which gives you the derivation path: Seed --seed-to-master-prvKey--> m/0h/0/0 (for the first receiving address) and m/0h/1/0 (for the first change address)

With the child private key of your (first) address derived, calculate its pair Public key in compressed format.
More info here: https://learnmeabitcoin.com/technical/keys/public-key/

With that compressed public key, encode the Native SegWit (bech32 - P2WPKH) address.
Bech32 encoded address step by step guide: https://bitcointalk.org/index.php?topic=4992632.0
Tool: https://learnmeabitcoin.com/technical/keys/address/#p2wpkh

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Note: Those are specific to the latest default "Standard" wallet, other wallet types including "old" and legacy p2pkh electrum wallet have different derivation path from m.