Thanks everyone for the speedy and informative replies. So, as far as I understand it, the process is as follows:
1. Apply Glyphosate weedkiller - wait until all grass/weeds are dead (takes about 2 weeks) and reapply to any spots, if necessary.
2. Rotivate the dead lawn.
3. Top up with topsoil to raise the level to required height.
4. Use roller to level and compact.
5. Lay new turf.
6. Keep watered until well established.
This can start when growing season is underway, so anytime from late April onwards.
Sounds like a plan; how hard can it be...