What is the length of time it takes for tb to show up in a person that have been infected?

To say that you are infected doesn't necessarily mean that
you have active disease. You could harbor the causitive pathogen Mycobacteria
tuberculosis in your body but not show signs and symptoms of the disease.
When this is the case you are considered a carrier.

On the
other hand, if you had the mycobacterium in your sputum a screening test used derived
from a sputum specimen called sputum for AFB (acid fast bacillus) named after the lab
test used called acid fast stain (a type of staining procedure in the lab) could show
the organism in the sputum within a couple of days after you were initially exposed to
and contracted the microbe. A sputum for AFB test does not mean you have active TB it is
an initial screening test. The sputum sample has to be cultured in the microbiology
lab.

People exposed to the microbe may or may not actually
get the disease and most do not. Many factors determine whether or not you will get
active disease, primarily the health of your immune
system.

A diagnosis of TB is only made when the persons
chest x-ray shows characteristic lesions. However, it may take months for you to have a
positive chest x-ray. The diagnosis may be R/O TB, rule out TB. If you are suspected of
having TB by showing clinical signs and symptoms like fever, chronic cough, night sweats
coupled with either a positive sputum AFB or positive PPD (skin test), the treatment is
the same regardless. Note that a positive PPD alone does not mean you have
TB.