Wet hair may make you feel cold, but it does not infect you. You need exposure to respiratory viruses or bacteria to get an RTI.
Cold air itself doesn't cause RTIs—these are triggered by viruses or bacteria. However, being in the cold for prolonged periods may slightly reduce immune response in the airways and can dry out the nasal passages, making it easier for infections to take hold.
Infection typically requires bacteria to enter the urethra, which doesn't happen just from sitting on a seat. Good hygiene is helpful, but toilet seats are rarely a source of infection1.
(1) Quote from Dr Jose Tiran, Gynecologist, Mexico.
Occasionally holding in urine is usually fine and won't cause a UTI. But regularly holding it for too long, it can let bacteria stay longer in your bladder, increasing the risk of urinary infections. UTIs are usually caused by bacteria getting into the urinary tract, not by one delayed bathroom trip1.
(1) Adatto et al. Behavioral factors and urinary tract infection. JAMA. 1979;241(23):2525-2526.