I just did this lubrication as my 2003 shifter would stick when pressed down. I used the urea grease from honda.

Some tips on doing the job:

If you pull the nylon sockets off the shifter there is an orientation to the bigger one:

The big nylon socket has a key in the front.

The (North American) shifter has a tab near the bottom that points to the left side of the car. The shifter top has a bend to the right side of the car. I'm not sure if UK cars are reversed.

Also I noticed that the big nylon socket would not fit easily into the machined housing. It would hang up and simply would not go in. I tried just fitting the nylon socket and the problem was clear. It seemed like the nylon expanded (maybe from the old grease chemically reacting or from heat or from use). I ended up using a file to lightly reduce the outer surface of the nylon socket so that it slid into the housing better.

I am also guessing the problem may not be drying grease but nylon expansion.

I hope this helps someone!