Do I need a transit visa to stay airside in Heathrow, as an Indian going to the Cayman Islands?

There's a handy Do You Need a UK Visa wizard on the UK Government website, which you can use to check. The answer for you is that you will need a transit visa even for airside transit. Indian citizens are one of a small number who need a visa for airside transit, except in a few cases:

You won’t need a visa if you:

• are travelling to Australia, Canada, New Zealand or the USA and have a valid visa for that country
• are travelling from Australia, Canada, New Zealand or the USA and it’s less than 6 months since you last entered that country with a valid entry visa
• have a valid US permanent residence card issued on or after 21 April 1998 (or an I-797 extension letter issued by the Bureau of Citizenship if it’s expired)
• have a valid Canadian permanent resident card issued on or after 28 June 2002
• have a valid uniform format category D visa for entry to a state in the European Economic Area (EEA)
• have a valid uniform format residence permit issued by an EEA state

Cayman Island issued visas sadly aren't one of the ones on the exclusions list, so you'll still need to apply for a Direct Airside Transit Visa before you can change planes in Heathrow (or any other UK airport)

(While the Cayman Islands are a British Overseas Territory, they have their own immigration rules, and issue their own visas and work permits, so their status doesn't directly help you)