Sir William Bradshaw is the character who most embodies British upper-class patriarchal oppression. Okay, that's a lot of words. Basically, he's British and proud of it (no exceptions); he's rich; and he thinks men (as opposed to women) should be in control. Professionally, he is a psychiatrist, and the public majorly respects him. So much so that he is never really second-guessed.
In his practice of medicine, Bradshaw preaches Proportion and worships Conversion, which is an abstract way of showing that Bradshaw is a scientist and authority figure more than a caretaker. Proportion is a way of expressing his belief that everything must be in order, that everything has to be balanced and in line, and that people must behave according to their place. Conversion reflects his belief in his power to change people and put them under his control – to "convert" them into obedient citizens and patients. He’s concerned with preserving the British way of life in which men are men (hoo-ah!) and suffering is done behind closed doors, if at all.
Bottom line: not a good guy. But he got to Septimus a little more than Clarissa (who also went to see him once); in the end, Septimus commits suicide to avoid going with Bradshaw to one of his psychiatric homes.