In the UK, permission for weather balloon flights is granted by the CAA, to get permission you need to fill in this CAA Permit Request at least 28 days in advance (even earlier if possible). To fill in the application you will need the latitude & longitude of the launch site, which you can get from Google Maps. Due to the changeable UK weather it is worth applying for a 'window' of possible launch times - such as a couple of weekends.
The CAA will send you a permit document if you have been granted permission to launch - it normally contains some form of co-ordination instructions with a local airfield that need to be followed before launch. You must be in receipt of a Permit before you can launch.
The actual situation is not that you are applying for permission but instead being granted exemption from the Air Navigation Order 2005 surrounding the launching of balloons. The regulations require the use of a standard meteorological balloon and for the payload to descend by parachute.
The most important thing when planning and launching high altitude balloons is to be safe. For the benefit of other flying vehicles, such as planes, when you get 'permission' from the CAA they publish a NOTAM which will alert pilots that you are launching in the area that they are flying in. Also just before you launch it'll be necessary to call the local Air Traffic Control tower to tell them that you are about to launch, and they may ask you to wait a bit if there are planes in the area.
Other methods of being safe include striving to make your payload as light as possible (which will save on helium costs and also get you just that little bit higher!), and insulating it well will help with the low temperatures but also help to cushion it on landing. As required by your 'permission' you'll need to attach a parachute to your payload to slow its descent.
Finally you should only launch when the winds are right - it is not advisable to fly when the forecasts have it landing in a densely populated area - for example when launching from Cambridge, UK we don't fly when the forecast has it landing within the M25 (i.e. close to London).
In the UK, OFCOM regulations stipulate that amateur radio cannot be used in airborne applications. As such, license free modules must be used - though not all license exempt modules are permitted to be used in air to ground communications. See license exempt for more information.
Regulations may vary in other countries - ie American and Australian legislation appears to allow the use of amateur radio transmitters in airborne applications. Don't take this as gospel - check with your local regulator.
Indeed the Radiometrix NTX2 does only transmit ~500m when on the ground, the limited range is due to poor line of sight (LOS). Attached to a balloon transmitting at a slow data rate and with a sensitive receiving antenna and radio the LOS is excellent and so it is possible to get over 400km range! Most commonly we use RTTY at 50 baud and use amateur radio receivers such as the FT-790r, FT-817 and Icom IC7000 and yagi antennas to receive.
Most of the discussion about HABing both in the UK and around the world takes place on the IRC channel - #highaltitude on irc.libera.chat - feel free to get on there and ask questions. There is also a mailing list which has quite low traffic but usually has the launch details of upcoming flights published. There is loads of information on the wiki so have a browse - if you find something missing you are very welcome to fill the gap! Finally check out the various projects, many of them have pages on the wiki and others have their own websites - they can be great guides and great inspiration for your project!