Is there a relationship between sciatica and urinary tract infections?

It is possible: If the nerve impingement from the herniated disc involves nerves that supply the urinary bladder, then a patient may have incomplete bladder emptying which could increse the chances of developing a uti. You should see your physician so that they can obtain a kidney and bladder ultrasound as well as refer you to a urologist for further evaluation.

No: Both could cause back pain but from different reason, most likely the pain from sciatica is in the lower part of the back, the other could involve the upper part of he back.

No, unrelated: Both are fairly common conditions & thus certainly could coexist. Also both can cause back pain, although kidney/uti pain tends to be located higher up at level of posterior rib cage. Sciatica is a set of symptoms (low back, buttock or leg pain + areas of skin numbness) caused by various conditions: spinal disc herniation (slipped disc), spinal misalignment, spinal stenosis or pyriformis syndrome.