An elbow dislocation
disrupts a lot of ligaments around the elbow. However, there are major blood vessels that cross the elbow, too, which can be pinched, bruised, or partially torn (called an "intimal tear" for the inner lining of the vessel), causing decreased blood flow to the hand. There are a lot of criss-crossing vessels (collateral circulation) to supplement bloodflow, so major injuries may not be obvious.
While some swelling and bruising are expected after an injury like this, your hand should have normal feeling and be warm like the other hand. If you check your pulses at each wrist, they should feel about the same. The swelling should go down with elevation
I assume you mean it has been a week since your injury. While a vascular injury is less likely to show up at this point, clots can still happen.
If you have any signs of poor circulation, go to the nearest emergency room! otherwise, call your physician to be seen today.