AC Electrical Safety Guidelines/ Legislation for in-house ATE
There are electrical safety standards that must be followed for your product as a whole. These are usually EU standards rather than UK standards. Unless there is a product specific standard you probably should use EN60950 and all of its amendments. I'm not aware that there is a specific standard for using a an AC/DC module, its use will be covered in the main standard.

It is difficult to give advise about the safety of your product without a sample of the product and a thorough knowledge of how it is intended to be used. And anyway this site is not meant for that. You asked a few specific questions:

Do we need to earth the chassis?

You should must earth the chassis of the AC/DC module. This is for both electrical safety and EMC. The earth connection is often used to absorb high frequency interference coming from the AC/DC module, and also to absorb high energy transients going to it. Generally you must also earth the metal chassis of your product.

Is an MCB required?

Usually, no. However, you should use a glass fuse in the Live conductor. Most people would use a Fused Terminal Block for the mains connection.