I have done head swaps successfully on two occasions, however never with anything important. For valuable data I would go with data recovery experts. I've done this on a ~200 MB 1990's Western Digital Caviar, and a 40 GB Western Digital, at home without specialized tools.
Doing a head swap requires having an identical hard drive available as a donor. I kept my work area clean, wore gloves, and gently dusted all parts with canned air to remove any dirt that may have settled before closing up.
For both the failed and donor drives, with tweezers I carefully lifted each head, then slipped PTFE tape underneath to allow the heads to slide along the disks for removal without damaging the platters or heads. The tape extends off the edge of the platter so the heads slide right off the platter and the opposing heads then press against each other with two layers of PTFE tape between them. This prevents the donor heads from smashing together, and allows them to easily separate for reinstallation.
Reinstalling the heads is essentially the same process in reverse; this step is tricky, especially so with multiple platters.
I later attempted recovery on a 2.5" laptop drive for a friend who decided it wasn't worth paying for proper data recovery, and he was willing to risk permanently losing his data plus destroying a new donor drive. I was much more careful this time, and even did the swap inside a clean room, but the smaller parts were very difficult to handle. I was unable to get the drive working (swapped PCBs first, then swapped heads and tried both PCBs).