To get there, I think you'd be best off on the tube rather than the Heathrow Express. It's not very quick from Paddington station.

By tube, you'd want to take the Piccadilly line to Hammersmith, cross platform change to District line to Blackfriars, then a couple of mins walk. I think that'll be quicker and easier than changing at Holborn for the central line, IIRC that's a bit more of a faff.

If you do end up on the HEX to Paddington, quickest (other than taxi) would be to walk for 5 minutes to Lancaster Gate (moderately well signed, fairly easy walk), then Central line direct to St Pauls

Either way, I think 1pm would be cutting it a bit fine. If everything goes well you'll make it, but you'll be stressed, and any delays means you won't