With supporters having a whip round to pay for damage to the AFC Telford fans' coaches that were wrecked last week, Kevin Wilkin's side trailed to goals from Tait and Brayson.
Spencer pulled one back for Nuneaton in the 51st minute and a minute later, Kyle Storer equalised. With six minutes of the match left, Moore hit the winner to move Nuneaton up to fourth place.
Former City youngster Ben Mackey scored for Corby Town in a 2-2 draw against Guiseley.
A crowd of 444 saw Leamington beat Tiverton Town 2-1 with a brace from Luke Corbett after a third minute Morley own goal had given the visitors the lead.
In the Central Division, Blythe scored twice for Bedworth United in a 2-2 draw at Ashford Town (Middlesex, Rugby Town drew 0-0 at home with Biggleswade but Atherstone Town were beaten 3-0 at home by Bedfont Town.
Stratford Town drew 2-2 at Studley Town in a Midland Alliance game.
In the Midland Combination Premier Division, Southam United won 1-0 at Castle Vale, Alveston beat Pelsall Villa 1-0, Nuneaton Griff beat Continental Star 2-1 but Racing Club Warwick lost 2-1 at home to Brocton.
In Division 3, Stratford Town A were beaten 5-1 by Coventry based Future Legends whilst in the Birmingham Junior Cup, Coventry Amatuers beat St Francis 2-1.