1. Were you able to observe how the deer reacted when hit?
2. When the deer ran and walked, did you observe any indications the deer was severely wounded?
3. Did you find and observe the spot the deer was hit?
4. Any blood of any amount, hair, meat, tallow, etc, at spot deer was hit?
3. Did you find your arrow?
4. Did you look for your arrow?
5. Where the deer was standing after being hit, were you able to determine if any of the blood was pooling more in one hoof print?
6. Was the blood dark red?

Hard to say how long you should wait being you are unable to provide much that would indicate how bad the deer was hit and that you have not found your arrow to help determine how bad the hit may have been.

I have seen deer that were dead on their feet just meander away like they had not been hit. However, it appears that you might have clipped the deer or hit it in a muscle or bone and/or did not get much penetration. Hard to say.

I rarely leave the area of a hit. However, a 2-3 hour wait sounds reasonable.

PS: Just a constructive piece of advice being it appears you might have put the cart in front of the horse. You need to know all that you can learn about what you do and what you look for after you hit a deer. That is when the real work begins; and having the necessary tracking knowledge and skills are critical.