Not sure about UK to Africa but interms of Tunisia to Sahara it depends on where you wish to go in the Sahara?
You can get into the desert in SW Tunisia quite easiliy when you are there, lots of tours do single day or multi day tours. You can also travel through the Sahara across to Morocco however this is much more specialised because you wish to travel through Algeria, which is only starting to deal with tourism.
You can travel east across Libya and into Egypt. this is much easier as there are tour companies that will take you through each part (however one that does this in a joined up way could be more difficult).
Can help more if you have a route in mind.