Between 5,000 and 6,000 years, using a whole lot of assumptions
For comparison, the American colonies were 2 to 3 months away from their mother country in Europe, and the Indian colonies about 4 months away. Using this comparison, it took between 150 and 300 years for those colonies, respectively, to break away from their original nation and establish themselves as independent.
Depending on your interpretation of modern politics, it could be claimed that we are closer now (in 2018) than ever before to establishing One World GovernmentTM, and we're still several decades off. Again looking at the current time period as the end of the colonial era, it could be said that it has/will have taken between 500 and 600 years for the following cycle to occur:
- Home nation begins acts of colonization
- Colonies established
- Colonies develop/become self sufficient
- Colonies revolt/declare independence
- Colonies form economic/social/political alliances with original nation
- Home nation and colonies reunify into one nation
Using this as a model, we could assume that a similar time scale might apply for extra-solar colonies as well, assuming that the travel and economic separation is similar, as you described it is. However, there is one key difference: colonies on Earth are/were limited in that a) there is a finite amount of space to go around and b) colonies very rarely, if ever, began establishing colonies of their own.
So while this 500-600 year cycle holds accurate for Earth, where colonizable area is finite, in the vastness of space with thousands, hundreds of thousands, potentially millions of planets to colonize, the cycle cannot be this fast. So instead, lets assume that the 500-600 year cycle refers to one "round" of colonization, and as a cycle completes the now-former colonies will begin colonizing anew of their own accord.
Modern estimates using the Drake Equation put the number of life-supporting planets in the Milky Way at around 15.6 Million. Let's just discount for the moment the possibility of terraforming, or colonizing non-life-supporting planets.
Assuming that each planet establishes an average 5 colonies per colonization cycle, and using the exponential growth formula, this means that all 15.6 million planets are colonized with 10 cycles, or 5,000 to 6,000 years.
All in all, it will depend on how your civilization works. If it is plagued by wars, it will take longer. If it is unified behind a progressive religion it might take shorter. However, assuming a social, economic, and political climate not that dissimilar to modern industrialism, this estimate may be accurate.
Two related references:
From Elite: Dangerous:
The galaxy's total NPC system population is approximately 6,632,313,733,791 NPCs on January 12, 3303.
The Core Systems also known as The Bubble is a sphere of about 150 light years around Sol with 19 thousand inhabited systems where powers compete for influence, resources, territory and control. The territory volume is about 13.8 million cubic light years, and encompasses over 20,000 inhabited systems with over 66,000 stations, starports and outposts.
Set about 1,300 years in the future, humanity encompasses a very small portion of the Milky Way
From Foundation by Isaac Asimov:
"(theatrically) Do you realize, Dr. Seldon, that you are speaking of an Empire that has stood for twelve thousand years, through all the vicissitudes of the generations, and which has behind it the good wishes and love of a quadrillion human beings?"
Set 12,000 years after any recognizable time, the Galactic Empire of Asimov's Foundation occupies the entire Milky Way, and has done so for so long that its scientists have forgotten what planet Humanity originated on