My father just started dialysis last week. His wearing diapers because his not able to go to the bathroom on time is that normal?

Kidney failure. The reason why your Dad has been put on dialysis is that his kidneys are no longer capable of filtering his blood well enough. When they can't do that various build up and will quickly kill him. They may still be able to generate a very small amount of urine but the kidneys do more than that. They help regulate blood pressure. The incontinance is not his major issue.
Incontinence . Urine incontinence is not due to his being on Dialysis but rather a bladder issue. Enlarged prostate can be an issue. Most patients that have stated dialysis and while in it can make urine in varying amounts. Suggest talk to his nephrologist about this and discuss options.