I had the same problem a few years ago. This was the solution.
I have just had the same problem on my 1.4 Greenline 08 plate. I took it to the local garage that I use for servicing and they suggested it was the gearchange cables that needed replacing. I decided to investigate on the internet myself and came across a number of videos on YouTube that suggested it was the gearchange relay shaft that was seizing up. This affects the left to right movement across the gate. I asked my son in law mechanic if he would take a look and this is what we found.
1. Disconnect battery.
2. Remove battery.
3. Remove air cleaner box to help with access.
4. Remove battery tray held on by 3 screws. This gives access to the gearchange linkages on top of the gearbox.
5. Disconnect the gearshift cable connectors which connect to the gearshift relay. On mine they just pushed on to the shaft although on some models there is a circlip holding it on. Do not move the springs on the end of the gear cables otherwise you will have to calibrate your gearchange afterwards. Try moving the gearchange from inside the car. If the cables are to blame it will feel stiff. Mine were free to move.
6. You should then be able to try and move the gearchange relay. It will be very stiff due to corrosion or dryness on the shaft. On some the shaft is metal and goes through plastic bushes at each end, and locked on the end by a circlip. I think mine has the updated one as there was no circlip on the end, just an expanding compression end which opens when it is in place. Rock the relay back and forth whilst compressing the end of the shaft with fine nosed pliers. The whole shaft is plastic and needs great care when forcing it out with a prybar but plenty of wd40 and gentle rocking back and forth will ease it out. Once out I could see that the plastic shaft was bone dry.
7. If you have any Emery tape just clean out the tunnel that the pivot goes through.
8. Lubricate the shaft of the relay with high melting point lithium grease. And push the relay back on. It should pop back in easily and move freely.
9. Pop the gearchange linkages back on. Test the gearchange from inside the car. It should be so much easier.
10. Put it all back together...simples.
This video shows what it looks like. It took us about 90 minutes in total and could easily be done by a non mechanic (me). This problem started after being parked for 2 weeks at the airport whilst on holiday.
https://www.briskoda.net/forums/topic/460703-gearstick-stuck-not-moving-left-or-right/?do=findComment&comment=5170825