Yes, it's important to remember on the UK rail network print at home really means what it says. It's not like a flight where you can pick up a real boarding pass even if you've already printed one.
If in doubt whether you'll be able to print always select collect at station, plus standard rail tickets are a nice credit card size not cumbersome bits of A4 paper.
Excellent work by the East Coast staff member who offered to print your tickets for you.
I hope you caught your Paddington train without issue.