If the penis urethra had a bacterial infection, would a quick urine dip stick test at the gp show positive leukocyte and nitrates?

Needs careful review: Urethritis can be 'obvious' as in the case of gonorrhea or less obvious in other cases -which are typically referred to as NGU - non-gonococcal urethritis. For a thorough evaluation, the doctor should go beyond a simple urinary dipstick. NGU will only have positive findings some of the time with that. I prefer a nucleic acid amplification test (NAAT) that would be much more sensitive.

Urethritis: There will be cases that every test will give a false negative result. These are because the most prevalent agents in this pathology (Gonorrhea and Chlamydia) require very special techniques and culture medium, in some cases they cant be isolated. So if your symptoms dictate it, you should receive an empyric treatment for these pathogens. I would recommend you to see an urologist.