I turns out that what I want to spend and what I have to spend are a fair bit apart.
I have found nothing off the shelf which comes close to fitting in the space I've got. In further review, the wheeled tool chests are too high and too narrow – not making good use of the space.
The only plastic trays I found (see above) are the about the right size for drawers, but the only web site that sells them (that I can find) doesn't work and in any event were going to cost quite a bit more than my final solution. In addition there is almost as much joinery in making the shelf unit they would need.
So. Despite everything, I have become a joiner again. The wood and runners are going to cost me around 100 quid which is more than I wanted to spend, but there you go.
So I expect to end up with 6 drawers, about 75mm hig0, 350mm across and about 540 deep. Good use of the space and perhaps I will learn some joinery. One drawer already made and looking better than I expected.
Thanks for the help and pointers.
I will post a picture when done if I'm not too embarrased.
Iain