There are two kinds of connections: "airside" and "groundside."

Airside connections are what you have if you're arriving by air, do not clear customs or legally enter the UK, and depart by air.

Groundside connections are if you clear customs, enter the UK, and re-enter the other terminal like any other departing passenger.

Airside, it's the shuttle bus. Period. These leave every 10 minutes from every terminal to every other one. The trip is about 5 minutes among 1-2-3, 10 minutes from any of these to 4. Your hand luggage will have to be re-Xrayed when you arrive at the other terminal but there is no passport control, no customs, and generally no lines (queues) to speak of. Follow the signs to Flight Connections. It can be a longish walk. Still, there is no reason it should take more than 45 minutes worst case, arrival jetway to departure gate. Add to this however much time you want to arrive before your flight leaves. You won't need much if you get your boarding pass at your originating airport.

Groundside, you walk among 1-2-3. The best way to get between 1-2-3 and 4 is the Heathrow Express train (the same one that goes to Paddington Station in London). It leaves every 15 minutes and takes 5 minutes for the (free) journey between terminals.

If you have no checked baggage and qualify for fast passport control (a Fast Track card, an EC passport or anything else) groundside connections can be faster, especially within the north (1-2-3) complex if you're a fast walker.