The fights starts at Naboo and Tattooine per the OP. The Jedi are based on Coruscant, and at that time, per the film, the only Sith were Palpatine and Maul (though Dooku was likely in the shadows, he didn't appear on the scene until Anakin was a teen, so several years later). Word of the problem would have to reach the Jedi council and they'd have to travel to those planets. By the time they arrived there'd be millions of Borg with hundreds to thousands of commandeered ships. It takes moments for Borg to be assimilated via the nanite injection tubules (as shown in TNG, Voyager, and First Contact). It does, however, take time for an assimilated Borg to be fully outfitted with implants (Picard was fully transformed into Locutus in about 30 minutes).
The Borg were assimilating the Enterprise itself in First Contact, a single Borg assimilated a pre-Federation freighter and outfitted it with advanced weaponry and upgraded propulsion in Enterprise, and they've assimilated numerous technologically advanced races in the Alpha and Delta Quadrant (among them the Krenim). Also Data is a very advanced android. He was built to mimic human sophistication, and there was even an episode where he and Geordi were running an experiment to use Data as a backup for the Enterprise's main computer (Fist Full of Datas), as well as his being fully inorganic. Borg assimilate races of humanoids and their tech (ships, devices, factories, etc). I don't think they'd bother assimilating battle droids when they can simply destroy them once their shields adapt to their blasters. It's the humanoid aliens and all of their ships that would have an issue. Depending on the ship, a Cube can have a compliment of a few hundred to a couple thousand drones. Battle Droids wouldn't be a threat after they encountered the first few Borg, fired on them, and the Borg's shielding adapted to their weaponry. Then it's a downhill battle for the Borg until some Jedi arrive on the scene and go all Bruce Lee on them.
I wonder if a light saber could be shielded against by drones? Jedi were unable to cut through shielding in the films. Force attacks might be the only real obstacle.