Under the Theft Act, body parts including blood, organs etc (but not sperm) are NOT classified as property.
Things change when there is an alteration process, such as embalming. Hence stealing body parts from a morgue, say, is not an offence under the theft act. Hence, if the morgue is broken in to for this purpose, it can not be a burglary.
An embalmed body, however, is property, belonging to the coroner.
Blood can be property if it has a plasma purification substance, for example, and it will belong to the organistion who has taken it. i.e the NHS, private hospitals etc, dieticians etc.
There is case law on this, but i'll be bu66ered if I can remember. The theives of unaltered blood (some vampire sect I believe) were charged with the theft of the containers or vials only.
Like all theft, the ownership of the "altered" blood can change hands many times. Once it is inside another human, the altered stae reverts back to unaltered. Live and connected body tissue and parts on a live person are not property.
There is also case law on somebody "stealing" hair by cutting it off. There was an assault, but no theft.
Hope this helps.