I arrive in Gatwick, south terminal on January 8 at 10:10am and I have exactly 3 hours to my next connection flight to Heathrow Airport terminal 3. Is this connection possible? What is the safest and best way of transportation to Heathrow?
You'd have to be extremely lucky to make this and I strongly recommend against this.
Some public options are here https://www.heathrow.com/transport-and-directions/travel-between-airports
Basically you can take a direct bus or go through London using the Gatwick Express to Victoria, tube to Paddington and Heathrow Express to Heathrow. The trains are most predictable the buses are subject to traffic which can be bad at this time of day. Google maps give the drive time anywhere between 45 minutes and 1 and a half hours.
Your best bet would probably be a taxi or a private car hire, but this would be very expensive and still no guarantee.
If you have checked luggage and/or need to go through immigration, this is probably a non-starter.
I checked the national express schedule. If you can make the 10:40am bus from Gatwick and traffic is not too bad, you get into Heathrow at 12:05. IF you already have checked in and your boarding pass in hand and no checked luggage, you may have a chance of making it, but it is very risky. You need to check the check-in and boarding cutoff time for your departing flight