Breaking the connecting flight

This answer assumes the OP still intends to travel to Atlanta, but with a few days stay in the UK.

If the OP does not change the itinerary in advance, there could be several problems.

With only a 3 hour layover at LHR, it is very unlikely that the the OP will be able to get bags checked to LHR. It will be very similar to a "hidden city" flight, and the usual advice is carry-on only.

The OP will still have to obtain a London->Atlanta ticket for the actual day of travel. The original one is likely to be cancelled when the OP fails to board. Buying the new ticket should be done in advance, so that the OP has an outgoing ticket if questioned by immigration, and for any opportunity for advance purchase prices. The new ticket should be completely separate from the old one, to reduce the risk of cancellation after the no-show.

If the OP arranges the stopover in advance, it should all go smoothly.

When the OP checks in at Mumbai, there will be no immediate connecting flight, so the bags will naturally be checked to LHR on the first leg. When going through immigration, the OP will have a visitor visa, a plan for a few days of tourism or similar in the UK, and a ticket, consistent with that plan, for the onward flight to Atlanta.

When the OP returns to LHR for the second leg, the bags will be checked to Atlanta.

There will be some additional cost, but it is likely to be less than the cost of a new London->Atlanta ticket, especially if the new ticket has to be purchased at full fare or close to it.

You can change your ticket now.

You have a one way ticket from Mumbai to Atlanta. All of BA's (public) one way fares are at least semi-flexible, meaning you can change them for a small fee. From what I can see, there is a 5000 INR penalty for changes on the cheaper tariff. The more expensive fares have no change penalty at all. I would advise that you make the change now.

You will also have to pay the UK air passenger duty if your new itinerary includes a stop in London over 24 hours.

If you are planning not to travel onto Atlanta, you may (or may not) be entitled to a refund on the fare difference between BOM-ATL and BOM-LON. Whether you are entitled to a refund depends on the exact fare you purchased. You may find it is cheaper simply to throw away the London-Atlanta leg without formally changing the ticket.

You can (probably) change your ticket later.

Even the cheapest public BA fare on BOM-ATL permits "no showing", admittedly at a fee of 10,000 INR, meaning you can simply not show up for the second flight, and still take it on another day. The more expensive fares allow no showing and rebooking without penalty. But I wouldn't advise it without checking, in case you are travelling on a private fare from a travel agent which has more stringent conditions. On the day of travel you would have to pay the difference up to the appropriate fare offered that day, assuming there were any seats available for you.

If you don't need to go to Atlanta, you can throw away the London-Atlanta leg and forget about it.

After passport control, you can go downstairs to baggage claim, tell them you have decided not to take your onward flight and you'd like your bags back.

In this circumstance, it may take a couple of hours for your bags to be found and returned to you.

As it currently stands, no. When you check in, you'll be given two boarding passes, one for BOM-LHR and one for LHR-ATL, but your luggage will be checked in for Mumbai.

Will BA be willing to replace your one ticket with two tickets covering the same legs, so you will be able to get your luggage at Heathrow? Certainly. Will they do it for free? you can ask, but airlines are famous for not doing things for free.

And three hours? If your plan is to get off the plan, go through Immigration, collect your bags, go through Customs, walk to Departures, check your bags, and get on the plane to Atlanta, three hours is just barely enough to do it safely.

If your plan is to do all that and take the Underground to Piccadilly Circus, turn around and come back, no, you don't have enough time.

If your plan is to actually see something on the island of Great Britain, other than transportation facilities.

EDIT: Patricia points out the OP is planning to extend his stay in Blighty. The airline will almost certainly charge for that.