Most biopsies of tissue go to a general Histology lab, there may be a few occasions where it will go directly to a specialist centre, such as IMF samples on skin, or in cases where lymphoma is suspected where the cells are studied, if you had a fluid sample taken, of say a cyst, fluid from your abdomen or chest, or knee joint where they are not querying an infection, it will go to cytology. If it is a fluid questioning an infection it should go to microbiology.
Your biopsies will have been placed in a fluid called formlin and allowed fix' for a minimum of 8 hours, before being described by a biomedical scientist, and then processed. Processing normally takes 13+ hours, and during it the water in your tissue is removed and replaced with wax. Once this is complete the tissue is placed into a wax block, cut, stained and given to a histopathologist. Who will look at it and ask for further stains (add another day) before reporting. If her2 testing isn't carried out in house the block will be sent away for staining before the slide is returned and viewed by the pathologist.
Generally labs say 7 working days to get a result, nothing about histology is fast. we can normally get breast cores on thursday, process friday night, cut and stain monday morning, pathologists look monday lunch or afternoon, ihc requests by 3:30 and have those stained for the MDT at 12 tuesday, but it's tight.
We like adequate biopsies, because if the first core missing the tumour you have gone through that for nothing and it will need to be repeated, we can do nothing about a diagnosis if we do not have the abnormal cells to look at. Also if there is not enough tissue (lung biospsies are the worst) we can't do all the testing and so the results will be incompleted and further tissue requested, especially in the era of genetic testing.