If someone's arm is cut off, how does that affect blood circulation throughout their body?
If I understand your question correctly, you ask how does the body deals with released blood volume after the removal of part of blood vessels?

The simple answer is the body decreases the blood volume. Our body is able to alter blood volume significantly in response to different factors.

For example, take a look at this paper: Blood volume changes in normal pregnancy. It states that: 


  A healthy woman bearing a normal sized fetus, with an average birth weight of about 3.3 kg, will increase her plasma volume by an average of about 1250 ml, a little under 50% of the average non-pregnant volume for white European women of about 2600 ml.


So, after the arm cutting, an organism will decrease its blood volume. Although it will take some time, blood volume altering is more probable way than new vessels generation.