There are many aspects to "time lapse construction cameras" to consider and many possible solutions.
Some of the "construction camera" vendors like OxBlue or those mentioned above use SLR cameras like Nikon or Canon, that take snapshots at timed intervals, send those snapshots via a cellular router to their servers/software where the end user accesses their time lapse videos. These SLR cameras typically deliver much higher image quality images compared to most IP cameras. However, these SLR cameras don't allow for live viewing, as far as I know. And the business model for some of these "construction camera" companies can include monthly charges of up to $500 per month per camera, for hosting images and providing the user interface, plus the cost of the equipment.
For a relatively low cost time lapse service, we have used TimeCam.TV. Their service starts at $20 per month, should work with any IP camera. Here's an example of a construction site that is using a Mobotix M15 6mp camera for viewing live, and TimeCam.TV for current time lapse videos. System is also sending 6mp images via FTP for custom time lapse videos. Here's an example of a custom time lapse video created by JDP Media, camera used was an older model Mobotix M12.
Other variations on "construction camera" requests can be time lapse videos AND/OR live camera viewing AND/OR full video security management features AND/OR a robust perimeter intrusion detection system that detects intrusions and allows for pro-active intervention tactics like sirens, strobes, recorded warning messages, talk-down through loudspeakers, live monitoring by a professional monitoring station, etc. These requests are all doable if the budget is there.
I've found that requests from end users for a "construction camera system", when questioned in detail about what they expect from the system, will often morph into some of these scenarios described above.
A couple of popular cellular router mfgs that will connect a camera to the internet are Sierra Wireless or Cradlepoint. For about $50 per year, Cradlepoint offers a fairly sophisticated cloud based dashboard that allow users to monitor all aspects of their cellular system, including analytics for performance, data usage, cost, etc.
final note: If there's no AC power available, which is common with "construction cameras" they can usually be powered with solar electric power.