It depends on a lot of things like the kind of accident, the wreckage state, accessibility etc.
First thing to consider is that in most cases, the wreckage is evidence- so the investigating authority will take control of it and release it only when the investigations are complete or has reached a stage where the wreckage is not needed anymore.
As far as air investigations are concerned, the first thing to do usually is to dispatch a team to the site. For example, NTSB has a 'go' team which is dispatched as soon as possible to the accident site. From NTSB site:
At the core of NTSB investigations is the "Go Team." The purpose of the Safety Board Go Team is simple and effective: Begin the investigation of a major accident at the accident scene, as quickly as possible, ...
During their time on the "duty" rotation, members must be reachable 24 hours a day by telephone at the office or at home, or by pager.
After a preliminary investigation, the team decides what to do with the wreckage- either these are returned to the owner or retained for further investigation, partially or in full. From NTSB Major Investigations Manual:
When the IIC and the group chairmen have determined that parts or all of the wreckage is no longer needed for investigative purposes, the IIC ..., in consultation with the OAS Director, will be responsible for preparing and signing Part I of the wreckage release form (NTSB form 6120.15). The form shall be executed by the NTSB representative and signed by someone acknowledging his/her receipt of the wreckage This is done in almost ll cases, except those where there is no accessibility to the wreckage, like he ocean floor.
Part II of the wreckage release form will include a detailed list of any parts or components of the wreckage that will be retained by the NTSB for further examination.
Note that the release of wreckage is decided by the investigating authority and it may be kept in storage, if necessary.
There should be no pressure to release all of the on-scene wreckage. Often it is better to arrange for wreckage removal and storage and to retain control of the wreckage in case there is a need to examine it later.
In some cases, the investigating authority will go to great lengths to transfer the wreckage to their place, so as to carry out full investigation. This happened in the case of TWA 800 investigation, where the wreckage was picked up from seafloor and reconstructed to investigate criminal activity.
Image from the NTSB aircraft accident report (NTSB/AAR-00/03) fig. 29, page 102, via Wikimedia Commons.
Another thing is that the disposal of wreckage is a costly business when its done. This is where the size and location of the wreckage comes in. Most of aircraft wreckage would be aluminum scrap, usually too expensive to move and too uneconomical to use elsewhere. Obviously there's a difference between a 747 wreckage downtown and a R-22 wreckage in a remote hill (I'm taking about wreckage disposal, not investigation here). The disposal is done either by the owner/insurer or the investigating authority, with the owner/insurer coughing up. From Singapore Investigation of Accidents Statute:
If a person to whom custody of the aircraft, parts, wreckage or contents is to be released refuses to take custody thereof or fails to take custody within a reasonable period, the aircraft, parts, wreckage or contents may be disposed of in such manner as the Chief Inspector considers fit.
The expenses incurred by the Chief Inspector in disposing of the aircraft, parts, wreckage or contents shall be borne by the owner or operator of the aircraft and be recoverable from either or both of hem.
Most other countries have similar rules. As for reusing the contents in the wrecked aircraft, this is usually not permitted (though it depends on the operator/regulator rules). In case it is permitted, the LRU logbook should contain details about it being in an accident and in general, they are sent to OEMs for tests and recertificaiton. Some companies do trade in salvaged aircraft.
And finally the 'stuff' is sometimes left to salvage for whoever wants it- it may be the most economical thing to do, after all.