Let me start by saying that time is running objectively slower if they are moving in a gravity field (be it artificial or real). Two objects in relative motion with constant velocity don't experience an absolute time difference. Only a relative one. The objects are symmetrically treated. In the twin paradox it is examined what happens if one of the two objects in constant relative motion suddenly is changing its motion to reach the other object. Depending on how they got their relative motion in the first place, the absolute times on both objects will differ.
Well, the question. Imagine the speed of light to be 1 m/s. And let's assume your entire body finds itself in a place where it resides in a somewhat static state. No external factors affect your life.
Let's look at your blood flow (or your whole body). Obviously, when you move one arm, it gets accelerated, so there is an artificial gravity operating on your arm. This means the time in your arm is not in sync with the time in the rest of your body (which means the time in your body and in your arm are running at a different pace). When your arm is moving at constant speed again, both times are in sync again, but the time in your arm runs behind the time in your body.
What does this mean for your bloodstream? During the acceleration of your arm, the time in your arm is running objectively slower. This means that less blood is streaming through your arm wrt to the blood running through your body (let's assume your body to be at rest; there is some resemblance here with the twin paradox).
What will happen with your blood? Actually, nothing. The blood may be moving slower through your arm, but the amount stays the same (it's somewhat similar to missing an arm). If you also start moving your other arm, your legs, and your head (all to and fro, though this will be very tough as the speed limit is 1 m/s and the mass of your arms, legs, and head will get enormous so you are bounded in your movements) again, nothing will happen. Although the speed of your blood is different due to the "to and fro" movements of your arms, legs, and head, the amount of blood stays the same in all parts of your body.
What does change is the aging of your arms, legs, and head (when you keep moving them in an accelerated way; the accelerated movements of your terms, legs, and head experience artificial gravity which is equivalent to real gravity and as you probably know, time slows down in a gravity field).
So they won't age (or at least much less than your torso) while your torso is aging relatively very fast. So your torso gets old while your arms, legs, and head will remain almost constant in age. Your torso gets wrinkled, old (your vital organs start to disfunction, etc.), while your arms, legs, and head remain young (your brain can send signals to your arms and legs through your fast-aging torso). Very disturbing! And according to your very slowly aging brain, this happens in a flash. What a trip! Phfffuuu...
So, luckily, the speed of light is not 1 m/s, but about 300 000 000 m/s!!!