This is a good question, but difficult to answer. The difficulty is in determining or even estimating how much blood flows to the fingertips at any given moment. And to my knowledge there have been no experiments to determine this.
When a drop of blood is collected in the medical laboratory using the finger stick technique, the blood is said to be of capillary origin. This means that the blood at the tip of the finger comes from the capillaries which arise from tiny arterioles, which are branches of the small digital arteries, in turn originating from larger arteries in the forearm, which come from even larger arteries in the upper arm, which arise from major arteries at the aortic arch which branch off of the aorta itself, the largest artery in the body.
It is intuitively apparent that blood flow in the aorta is massive relative to the flow from small artery branches in the fingers, let alone from minute capillaries.
There are many studies that have shown blood flow in the major vessels (the aorta which leaves the heart, and the vena cavae which return blood to the heart). This amount of blood flow is roughly equal to the cardiac output. The cardiac output is the volume of blood pumped by the heart per unit time. In normal adults this figure is between 4 and 8 Liters/Minute, depending on activity of the patient.
We know that blood flow to the fingertips would be much, much less than the cardiac output, probably on the order of cc’s per minute. It could be estimated very roughly by an experiment in which a fingertip were slashed and allowed to bleed into a graduated vial (not an experiment I would recommend). Another approach would be to perform Doppler studies of blood flow in the upper extremities down to the fingertips.
Fingertip blood flow will vary with the cardiac output (which varies with exertion), and would be increased with the hand in warm water, and decreased in ice water due to vasodilatation and vasoconstriction, respectively. Vasodilatation (relaxation and distension of vessels) causes increased flow. Vasoconstriction (contraction and reduction of vessel diameter) causes reduced flow.