I terms of driving distance, you are looking at around 167 miles and about a 3 hour drive.
It takes roughly between 2.5 - 2.75 hours by train (did it yesterday!!)
It is fairly far from newcastle although it is slightly nearer to the north of england and is only about a three hour drive, but if you are driving from nottingham to Sheffield and south yorkshire it won't take long at all. But it is definitely much further from newcastle and the borders. But London and the south coast is definitely further away, but it's hard to tell, it's very much centrally located, not like Northampton which is quite close to eastern england and the southeast, but is still classed as the east midlands, I don't know why. But keep in mind that England isn't a massive country and it depends which area you area you are going from nottingham and which area you are getting to newcastle upon tyne too.