Versions
FF v48
Selenium Webdriver v2.53.1
Windows 7 (x64)
.Net 4.0 and .Net 4.5
Trying the fixes
As per @jaffa I manually added the following refereces and got the problem:
WebDriver.dll v2.53.1
WebDriver.Support.dll v2.53.1
I then unreferenced them and followed @HassanRahman and used NuGet:
WebDriver.dll v2.53.1
WebDriver.Support.dll v2.53.1
Still same error.
OpenQA.Selenium.WebDriverException : Failed to start up socket within 45000
Troubleshooting
I had a hunch and decided to downgrade from targeting .Net 4.5 to .Net 4.0 (Full not client profile) and the error changed to:
An unhandled exception of type 'OpenQA.Selenium.WebDriverException' occurred in WebDriver.dll
Additional information: Cannot find Firefox binary in PATH or default install locations. Make sure Firefox is installed. OS appears to be: Vista
I wondered if this is why people who reinstalled FireFox got it to work again?
Somewhere there has to be a set of Registry Keys or an Environment Variable to find FireFox's ApplicationStartupPath. You can see some of those paths in this ProcessMonitor trace:

In the screenshot above you can see the app (AstrobLabe) I'm using running this line of code:
using (IWebDriver driver = new FirefoxDriver(ffprofile))
It causes a search for FireFox v48 and its looking in the wrong directory:
C:\Program Files (x86)\Mozilla FireFox\FireFox.exe
On my system FireFox is installed in:
C:\Program Files (x86)\Mozilla\Firefox\FireFox.exe
Anyway finding and correcting this issue (Cannot find Firefox binary in PATH or default install locations) is for someone else to fix, I just copied the Firefox directory to Mozilla FireFox.
But then it goes back to the original error:
An unhandled exception of type 'OpenQA.Selenium.WebDriverException' occurred in WebDriver.dll
Additional information: Failed to start up socket within 45000 milliseconds. Attempted to connect to the following addresses: 127.0.0.1:7055
Solution
Then I came across this answer which says that FireFox v46 works with Selenium 2.53:
Which Firefox version is compatible with Selenium 2.53.0?
When I downgraded I saw that FFv46 installs to:
C:\Program Files (x86)\Mozilla FireFox\FireFox.exe
So check if your FF install is in this location - which based on the ProcessMonitor Trace Selenium cant find:
C:\Program Files (x86)\Mozilla\Firefox\FireFox.exe