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UK visas in India usually take hardly a week to process. 15 days is usually mentioned so that consulates are not blamed for delays. The UK High commission would be aware of your departure date and normally ensure you get your visa before that. You might like to track the courier to ensure they deliver your passport in time, after they receive it.
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My worry is, if I were to get my Visa on the 24th, would I still be able to board my plane?
Yes theoretically, as long as you have your passport with a valid visa at the time of check-in it does not matter when you received that visa. All that matters is that your visa should be valid on the date you fly in and on the date you fly out of the UK.
the date of my departure is the 24th of May (7:00 a.m. IST)
No in this specific case. For a 7 AM departure, you will have to reach there at least by 6 AM, and by 6 AM you won't be able to receive your passport either via mail or via the visa centre. You can only wait until the end of working hours on 23rd, don't count 24th.
Visa might be late
Visa might also be refused, have you planned for that?
You will probably receive a decision in less than 15 days, it can also be a refusal unfortunately. Whether you are willing to take such a risk is highly subjective. Some people will, some won't.
There are 3 possible outcomes of this siatuation
UKVI strongly advises against buying a ticket before you get your visa.