With cash or a wifi device, it's quite simple to make phone calls with public telephones or otherwise in the UK.
There are many public ways of making phone calls to India and the UK without a SIM:
Using free wifi, calls will be either free (WhatsApp, etc.) or cheap (Skype, etc.) but the connection quality will vary and you will sometimes have to buy a coffee or something else.
Using paid wifi, you will pay usually £5-£10/month plus the costs for wifi above.
Using public free or paid wifi or a phone box can waste a lot of time finding a reliable hotspot especially if you need to contact a friend to make arrangements. Also, if you are only in the country for a relatively short amount of time it might waste a lot of time.
Consider that you want to make a 15 minute call to India or a 5 minute call to a friend in the UK.
Using cafe or pub wifi, a coffee or pint will cost you £2-£6 each time but you will be able to make the calls for almost free.
Using paid wifi you'll have to pay about £10/month but you'll have to find the hotspots.
Using phone boxes, the 15 minute call will cost you £8.10 and the call to your friend will cost you
Another way which you have said you want to avoid but may well be easier would be to buy a local SIM but this is my recommendation.
If you buy a local PAYG SIM card you will save a huge amount of money if you are on a budget. Avoid MVNOs targeted at tourists and immigrants such as Lycamobile and Lebara as their rates are very uncompetitive.
This will cost about £10 and comes with cheap calls to UK numbers from as little as 3p/minute and 1p/MB or a month with 300 minutes and 500 MB of data for £6. Look into 3 Mobile for PAYG or Giffgaff for a cheap bundle that will give you enough data to make all you calls by WhatsApp or included minutes.