The most common cause of this type of problem is a worn oil pump in the transmission. An automatic works on hydraulic pressure and if the pump is worn, it often can’t muster the necessary pressure to effect a gearshift when it’s cold. Once it’s warmed up, everything moves a bit more freely and the car will select Reverse. The fact that there’s still a gear-selection delay even when warmed up is the giveaway that this is the case.
If you continue to drive it like this, it will possibly go for some months. But you need to be aware that the faulty oil pump is not only making shifting difficult, it’s also potentially starving the bearings of the lubrication they need. The end is nigh, I’m afraid. You could take a punt and have the gearbox serviced with new fluid and a clean filter, but, on balance, the pump is likely to be the culprit and is also indicative of general transmission wear and tear
