Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa?

2.8. Can a consulate accept an application from an applicant not residing in the jurisdiction of the consulate?

Legal basis: Visa Code, Article 6 As a general rule, only applications from persons who reside legally in the jurisdiction of the competent consulate (as described in points 2.1-2.5) should be accepted. However, an application may be accepted from a person legally present – but not residing - in the jurisdiction of the consulate where the application is submitted, if he can justify why the application could not be lodged at a consulate in his place of residence. It is for the consulate to appreciate whether the justification presented by the applicant is acceptable. "Non-residing applicant" means an applicant who resides elsewhere but is legally present within the jurisdiction of the consulate where he submits the application. "Legally present" means that the applicant is entitled to stay temporarily in the jurisdiction on the basis of the legislation of the third country where he is present either for a short stay or EN 24 EN when he is allowed to stay for a longer period of time while maintaining his permanent residence in another third country.

https://ec.europa.eu/home-affairs/sites/homeaffairs/files/policies/borders/docs/c_2010_1620_en.pdf

Usually no, you won't be able to apply for a Schengen visa in the UK.

The general rule, defined in article 6 of the Schengen visa code is that

1. An application shall be examined and decided on by the consulate of the competent Member State in whose jurisdiction the applicant legally resides.

And you're obviously not a resident in the UK so you should not apply there. The same article does however open another possibility, namely that

2. A consulate of the competent Member State shall examine and decide on an application lodged by a third-country national legally present but not residing in its jurisdiction, if the applicant has provided justification for lodging the application at that consulate.

If you just want to apply because it's more convenient for you, I think you can just forget about it. Beside exceptional circumstances the only thing I can think of is to argue that it was too early to apply in India (you are not supposed to apply more than 3 months in advance for a Schengen visa and you intend to stay 5 months in the UK).

But even if you do have a convincing justification, most consulates in London have stricter requirements. For example, here is what's on the website of the French Consulate General in London:

• Your UK residency must have at least 3 months validity after the intended date of departure from the Schengen area.
• As per the decree 2008-1176 of the 13th of November 2008, the French Consulate, London, is not competent in processing visa applications of people not resident in the UK, i.e. holders of ’C-Visit’ visas or of a British visa valid for 6 months or less. People who are not resident in the UK must apply for their Schengen visas at the French Consulate in their country of residence.

So even though they could (or perhaps even should) entertain the possibility that you have a good reason to apply in London without being a resident, they are obviously not very friendly to the idea. (I haven't checked all 20+ consulates in London but I came across similar language from at least half-a-dozen of them in the past.)

Furthermore, nearly all Schengen consulates in London use some third-party outsourcing company to filter visa applications. They will just go through their checklist, notice that you don't have the right UK visa and reject your application (this means they will decline to process it, it does not count as a refusal and there will be no traces of this). Merely getting to talk to an actual consular officer who knows the rules and could accept to examine it will probably be an uphill battle.

If at all possible, I would therefore apply in India (or in your current country of residence, if it's not India).

If you positively can't do that and still want to try your luck in London, I would try to take a print out of the relevant parts of the regulation and use it to insist (calmly but firmly) that your application should be examined nonetheless. Also join a short letter for the consulate itself in which you can show that you know the rules (mention article 6(2)) and lay out your argumentation.