A Time Lord cannot "live billions of years in 5 seconds." A Time Lord might move between two points in time billions of years apart, but they don't "get older" doing so. Time Lords age the same way everyone else does: how many seconds/minutes/etc their body has spent being alive. Since they do not travel linearly through time, you can't compute their age by simple subtraction, but they still age just the same as we do. For example, if The Doctor starts out in the year 2000, gets in his TARDIS, travels for 5 minutes, and exits in the year 10000, he's not 8000 years older -- he's 5 minutes older.
When a Time Lord gives an age, they're just telling you what their internal biological clock is saying about their age. When The Doctor says he's 1200 years old, he means "I have lived through 1200 years worth of time since I was born".