The talkSPORT Breakfast host was in attendance for the Magpies' 2-0 Carabao Cup final defeat to Manchester United on Sunday afternoon.
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Woods, joined by her Newcastle-supporting father and brother, enjoyed a pre-match meal amongst ex-stars, including Alan Shearer and John Beresford.
She also bumped into Taylor ahead of kick-off, but didn't initially realise who she was speaking to.
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Speaking to co-host Ally McCoist, she recalled: "I had a very awkward moment, can I quickly tell you about it?
"Ryan - I hope he's not listening yet - walked up and, just because we were frantically trying to find [our restaurant]. You know how big Wembley is and if you get lost, you're lost for days.
"We had been saying hello to all the Geordies and fans outside and then we got up and Ryan came up to me and said 'Hi, how are you?'
"I said 'Good thank you, how are you?', and he went, 'Yeah, nice to properly actually meet you in real life'. I went 'Sorry, who are you?', I didn't recognise him!"
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"No you never, man," a disbelieving McCoist asked.
Woods, reliving the moment, continued: "He went 'It's Ryan', and I went 'Oh God, Ryan, I'm so sorry!' It was awful!"
"McCoist jokingly responded that their talkSPORT show had just lost any chance of ever getting hi on, with Woods explaining she tried to make amends by asking if he'd join them on air if Newcastle won.
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"I'm so sorry, Ryan. I did spend the rest of the day apologising profusely to him."
On the pitch, though, there were no smiles at full-time as Newcastle were unable to leave Wembley victorious.
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First-half goals from Casemiro and Marcus Rashford were enough for United to lift their first trophy since 2017.
Magpies battled hard throughout the tie, but will have to wait for their next chance for silverware as they continue their impressive rise under Eddie Howe.