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A company called postcodeanywhere offer this service - paid but cheap. http://www.postcodeanywhere.co.uk/route-planner-services/
I have heard bing may offer a free service if you register for a key, but I have not checked.
You can do this from scratch wholy with filemaker with fairly basic maths provided you have the lat long / grid ref of each point and merely wish to use the distance apart - I and several other UK developers built means of doing this about 10 years ago.
This service has been built into our Deskspace Classic product since about 2006.
However, the biggest challenge is that the Post Office will want you to license the data - in many other countries this is public and free but here they have made it a profit centre.
This then constrains how the internet map companies operate as their licence from the Post office restricts their ability to give us this information - at least this was the situation a few years ago and I am aware of it having changed.
So you will need to either assemble a suitable dbase of post codes - if you just work off post code areas rather than discrete post codes you can find that data online - or license it and do the maths unless you buy the information.
Cheers - Nick