And since the UK requires a different set of tricks than the instructions I posted in January 2014, I thought it might be helpful to write a new post just for those with an Amazon.co.uk account so I can share the bad news.
At this time I do not know of a way for anyone with an Amazon.co.uk account to stream Instant Video on their Android device.
There's a trick where you can use Adobe Flash and the Dolphin web browser to get the service working on most Android tablets, but unfortunately it won't work with Amazon's UK website. The trick requires Flash, and Amazon.co.uk does not support that standard.
I can't tell you why there's no Flash support, but I can tell you that numerous comments have already been left by frustrated readers who have confirmed the bad news.
All I can say at this point is that I looked for a solution and did not find it. If one turns up in the future I will update this post.
P.S. In the absence of Flash, our only option is to use Microsoft's Silverlight standard. Unfortunately, Android is not supported - so far as I know. If this changes I will update this post.