Does mild hyperbaric oxygen therapy helps injuries heal quickly?

Possibly yes. The usefulness of hyperbaric oxygen therapy in healing chronic diabetic foot ulcer is well documented. There is only anecdotal evidence that hbo will help acute injuries heal. Inflammation is evident both in chronic wounds and in acute injury. Hbo is known to decrease inflammation so theoretically acute injuries should resolve more quickly with hbo. More study is needed before hbo use for injury.
"Mild HBOT" ??? It's my understanding that the term“mild hyperbaric oxygen therapy” (hbot) is sometimes used to refer to administration of 100% oxygen at less than 1.5 ata (atmospheres of pressure). It is not recognized as hbot by some, who may refer to this as “compressed air therapy”. Delivery of less than 100% oxygen under pressure may be reportedly called “hyperbaric” therapy. This info is from greg harris.
Mild? Injuries? I'm not sure what you mean by "injury" or your definition of "mild". For chronic wounds, radiation skin injuries, and known conditions, hbot only works at pressures above 2 atmospheres. While many sports teams have taken to putting their athletes in hyperbaric chambers, to my knowledge, there is no definitive proof that hbot has any beneficial effect on ligament injuries or muscle strains.
Hyperbaric oxygen. Your question is too general. Hyperbaric oxygen is used for several indications, some of which include injuries. For example, acute crush injuries where the flow of blood is compromised after an accident is one indication for hyperbaric oxygen. In this case, it is used to compensate for acute blood supply loss to a local area and is critical in the recovery of a limb.
Maybe, but... There is a big difference between "mild hyperbaric oxygen therapy" and true hbot. Mild chambers are not approved for use with oxygen. If used in this way, the oxygen content is much smaller and the pressures are lower than in true hbot chambers. True hbot does help injuries of all types heal more quickly.