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Captain Joel Appiah gave the fifth-placed Foxes a fourth-minute lead against their struggling neighbours.
But Woods hit back through Angus Hastings just past the hour mark as it ended all square.
Ilford were due to host White Ensign in the Memorial Trophy in midweek and visit Halstead Town on Saturday.
Woodford, meanwhile, had a home date with Soul Tower Hamlets in the trophy and make the short trip to the Old Spotted Dog ground in Forest Gate to face Clapton Community this weekend.
They welcome Romford to Ashtons next Wednesday (November 18, 7.45pm).
High-flying Frenford also had to settle for a point after a 2-2 draw with Harwich & Parkeston.
The second-placed side got off to a great start as Steven Carvell netted a ninth-minute penalty and Mo Malik made it 2-0 moments later.
But the Shrimpers replied from the spot before the break and snatched a draw in the last minute of normal time.
Frenford faced a Challenge Cup tie at Clapton Community in midweek, then visit Romford on Saturday and host White Ensign on Tuesday (November 17).