Up until fairly recently, I ran a large railway station and had to answer questions about left luggage every day.
Transec (the authority responsible for security on our railways) only permit left luggage facilities at stations where items can either be scanned by X-ray equipment or hand-searched by staff. For this reason, you won't find any lockers. As Dzug states, you can find left luggage facilities at most mainline termini in London. However, because these are all staffed offices, they're not cheap. (The facilities are provided by independent firms, not by Network Rail or by the train operating companies, so prices can vary from one station to another). Expect to pay around �6.50 per item, per day.
You can check the availability of station facilities, including left luggage, for any National Rail station, here:
http://www.nationalrail.co.uk/stations_destina tions/
At Paddington, the left luggage office is on Platform 12. It's open from 0700 to 2300 daily. To check the current prices, use the contact information, here:
http://www.excess-baggage.com/station_contacts .php#paddington
Chris