Start here: what does your own burns service already provide?
Before any conversation about private cost, it is worth confirming what your own health system already provides, since burn reconstruction, as treatment of an injury rather than an elective cosmetic choice, is usually funded in developed health systems as part of a burns service.
In the UK, burns services run through the NHS and typically include not only the surgery but also the pressure garment fitting, scar therapy and psychological support that determine how well a reconstruction actually turns out. Many other countries fund burn reconstruction on a similar basis, recognising that the garments and therapy matter as much as, or more than, the surgery itself. Where this funded combination is available to you, it is generally the more complete option, because it bundles the surgery with the non-surgical care that this page explains below actually does most of the work.