This is explained in the script. It's a semi-nonsensical phrase used to identify members of the 'Underland Underground Resistance' to each other.
MAD HATTER (cont'd): Have you any idea why a raven is like a writing
desk?
Hare, Hatter and Dormouse clasp their hands high.
HATTER/HARE/DORMOUSE: Downal wyth Bluddy Behg Hid!
ALICE: Sorry?
CHESHIRE CAT: "Down with the Bloody Big Head". Bloody Big Head being
the Red Queen.
THE DORMOUSE: It's a secret language used by us...the Underland
Underground Resistance!
and
INT. THE TEAPOT - ALICE'S POV
Alice peeks out the spout. Hare, Hatter and Dormouse sing on despite being hit, choked and dunked.
HARE/HATTER/DORMOUSE: Twinkle Twinkle Twinkle Twinkle!
The Bloodhound puts its paws on the table, sniffing Alice's teapot. The Hatter takes a chance. He whispers to him.
MAD HATTER: Downal with Bloody Behg Hid.
The Bloodhound stops dead. Hatter shoots a look at the teapot. The Bloodhound drops to the ground and sniffs, pretending to catch a new trail.
KNAVE OF HEARTS: Follow the Bloodhound.