Does my passport go to a foreign country to get the visa or is it issued in the consulate of ones own country?

The entry clearance officers are in the British embassy and consulates (known as high commission and deputy high commission in India). Your passport and supporting documents will be sent to the high commission or deputy high commission serving as a decision making hub for the area in which you live (see the map below), and an entry clearance officer there will process your visa application.

The amount of time it takes to process your visa can vary. It depends on a wide variety of factors. For example, the ECO or an assistant will verify your documents, and this can go quickly or slowly. You can look up processing times on the gov.uk web site to get an idea of how long it is likely to take.

The speed with which your documents are returned to you (or when you can collect your documents from VFS) depends to an extent on the service level you selected from VFS. Note that your documents may arrive separately from the letter giving the decision about your visa.

Overall, I would say that you have very little time. It can take 10-15 working days (and sometimes longer) for a visa application. If your case is very straightforward and all your documents are easily verifiable, it could go quickly. But, because you have already submitted the application, there is really nothing to do but wait.

In future, you should apply for visas as early as possible. You can apply for a UK visa as early as three months before your planned visit.

For reference purposes, the decision-making hubs and their relationships to the various application centres are shown in UKVI's graphic below...