I'm assuming that you mean Freesat and not Freeview (which is solely a terrestrial service).
Whether you're watching with Freesat, 'FreeSat from Sky' (which, confusingly, isn't the same as Freesat!) or Sky, the signals are exactly the same ones. (They're sent to your dish from a group of satellites known collectively, from their location, as '28 degrees east'). So it would seem reasonable to assume that you'd get good reception with Freesat HD.
However there has to be a small caveat in that different makes and models of equipment for receiving radio signals (including satellite boxes) have differing 'sensitivities' (i.e. their ability to handle weak signals). It's possible, though unlikely, that the box you purchase might have a slightly lower sensitivity than your existing one which (if the signal is very weak) could cause problems.
This map shows the footprint of the Astra 2D satellite, which carries (among others) the BBC channels, including the HD ones, as well as ITV1 HD:
http://www.digitalsat.../astra2dfootprint.gif
You'll see that there's a strong signal over much of France, Belgium and the Netherlands but that it rapidly drops outside of those areas.
Channel 4 HD (along with other channels) is transmitted from the Astra 1N satellite, with this footprint:
http://farm5.staticfl...0339_154a2b494f_b.jpg
NKH World HD (along with many other channels) comes from Eurobird 1:
http://www.vowles-hom.../images/eurobird1.gif
A few free channels are on Astra 2A South:
http://www.outdoorbit...tra_2A_2B_south_L.jpg
or on Astra 2B South:
http://www.lyngsat-ma...ges/astra2b_south.gif
You need to check your location against each of those maps to ascertain the degree of confidence you should have as to whether you can assume that all makes of satellite box will have sufficient sensitivity to provide reliable reception.
Chris