F1 live - BBC iPlayer - free
BTCC live - ITV.com/itv4 (not ITVplayer) - free
Other stuff live - Eurosport - £3.99 a month, no minimum term, so you can buy just one month if you want - and their accounts admin is so bad that more often than not you'll get 2 or more months free before they realise and cut you off. I bought a month's subscription in April to watch the cycling, cancelled the rebill immediately and could still access the site until 2 weeks ago. Not bad for £3.99!. It's best to watch the Eurosport HD stream which takes the European feed with an English commentary, although be warned that it's bandwidth hungry. If possible avoid British Eurosport, which, if their cycling coverage is anything to go by, is spoiled by talking heads in a studio before and immediately after the event. The European feed usually has a crew on site at the event for pre and post race interviews etc.