It is possible, but the preferred place is the home country. The second best option is the country of last residence.
There is no harm in applying for a visitor visa at a U.S. Consulate in the U.K. At worst, the Consulate will not issue it and indicate that your mother should apply in India.
You should visit the U.S. Embassy website in Scotland and/or UK regarding applying for visas.
Mr. Asatrian's practice is dedicated to the area of immigration and nationality law. Please note the information provided herein does not constitute legal advice and should not be construed as such by anyone. It should not be relied upon as legal advice as more specific facts, research and analysis may be required to formulate a proper strategy and action in your matter. Please note this does not create an attorney-client relationship whatsoever. You should seek the assistance of an experienced immigration attorney to review your matter thoroughly and gather all of the necessary information and documentation to provide you with the best possible legal solution.
Yes, this is certainly possible.
Disclaimer: This answer is for informational purposes and does not take the place of a consultation with an experienced immigration attorney. This answer does not create an attorney-client relationship.
The family members may apply for a solely discretionary non immigrant visa at the closest U.S. Consulate Visa Section on merits, however, the applicants must satisfy requirements of close ties to the homeland country, India in this case, and financial sustainability not to trigger a preconceived immigration intent.
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