Hi Richard.... you'll get many differing opinions on what to do and what to spend and what's best etc, so look and learn and pick up/ apply what fits to your car and use.
Area meets are a good way to pick up 'top tips' and passenger rides (knowing how the cars differ) will help inform you as to your preferred upgrade route.
"- replace all the tyres. i suspect the current tyres are the originals"
Replacing tyres and a full 4 wheel suspension set up & geo IMO should be your first upgrade, If the tyres are old they'll have hardened and not grip (Agree with Martin - I have Toyo R1Rs and think they're excellent for road and track days). Suspension and 4 wheel Geo will make a massive difference to how the Westie behaves, especially if done by someone who really know Westfields. Your local area members will know best places to get this done.
"- replace the brakes. most likely the pads..."
Westies do not have brake survos to assist with pedal pressure, so after a regular saloon may feel like not working as well. However after getting used to appying more pedal pressure they are actually extremely good. I changed to Mintex 1144 pads and did find they improved pedal feel and braking.
"- finally on the shopping list is a limited slip diff..."
As mentioned this is a costly upgrade and IMO probably not of benefit to your car and use. In fact an LSD could give you less confidence on road depending on what you fit and how it works, due to the potential greater tendency to oversteer when it 'locks out' on a corner. re non LSD "the one with the grip gets it all" I don't thinks quite correct - without LSD when there is an imbalance of grip the diff cuts the power to the wheel with grip. Balancing throttle application i.e. not putting foot through floor, goes a long way to eliminating this. If you feel you are having issues with 1 wheel spinning now it may be sorted by tyres, suspension and geo. e.g. I have a 155bhp zetec Westie (for 5yrs ish) and by far best upgrades were protec shocks, suspension/geo set up and driver Instruction. I do over 3000 miles a year on road, do track days and WSCC Speed Series Sprint competition and the only time I've felt potential need for a Limited Slip Diff is on a wet start line when sprinting, but compensate with careful clutch throttle balance and have achieved some better starts than LSD cars. I do believe that the need for a LSD may increase as engine power gets much greater.
Hope that is some good food for thought and helps
I'm sure someone will be along with differing thoughts soon
Enjoy