I don't know of any good places in particular, however i would not rule out trying for a sponsorship, my friends brother is in the very final selection process for a place at Flybe. In airlines it sometimes doesn't matter if you have gone to university or not. For example, my friends brother used to work as a tesco delivery guy for a short time. However unfortunately some idiots came along and forced him to into a ditch where they eventually burned the van (fortunately he was unhurt). This showed the airlines that he could cope with being under pressure, a useful skill if you have to deal with a big problem while your flying. As far as i know i think there is around 7 places for the sponsorship my friends brother is in. I'll try and let you know how he does.
The blood test only needs a small drop of blood. They use a thing that goes "clik" to prick the end of your finger. It is only a little painful.
As for pilot training, I suggest you first get A level Maths and Physics though that's not mandatory. Mostly you need money. If your dad is rich ask him to pay about Â£100,000 for your training to include Frozen ATPL, MCC and a type rating on a popular aircraft such as Boeing 737 NG or Airbus A320. Then you will have a good chance of getting a airline job.
If your dad is poor, your chances of becomming a airline pilot are not good unless you are very determined. The vast majority of airline pilots in the UK are white middle class men. The airlines are not politically corrrect and are not interested in "widening access" to the pliot profession for "under represented groups".
First steps? Go to/ dont go to a university. If you dont, you cannot fly for major airlines like BA. Find a suitable flight school. Check out what licenses they offer. Theres a students pilots license, a private, and commercial. Then they are instrument ratings and multi-engine ratings and a flight instructor license. You need these licenses to become a pilot, so make sure to pick a school that offers them.
The sponsorship? I might not even bother looking for one, you need to have really good test results and to have gone to a university. Even then its only like 5% of people who apply for them that get them. And anyway airlines aren't offering them now because of the recession.
The medicals... I would think that they would use needles for the blood test, sorry! But you shouldnt let that fear get in the way of a great career.