Below is a list of all the suppliers who I have found that will ship to the UK, or are UK-based for bento boxes and accessories.

Not on the high street
Urban Outfitters
Marks and Spencer
Squeaky Bee
Asda
Muji
Artbox
Paperchase
Gifted Penguin
The Holding Company
Japan Centre
Tokyo Jazz Panda
Ebay
Dot Com Gift Shop
Casa Bento
Bento and Co (Now available in English!)
J List
Girls travel club – 1 bento box in stock
Lakeland Limited – for lock and lock
QVC – for lock and lock
Foodnut Australia – Lock and Lock Lunch Kits

Fabulicious – For thermal lunch bags
Amazon UK for lunch boxes and bags (Amazon are now UK supplier for Laptop Lunches)
Modes 4 U
Japanese Kitchen For decorative laquerware bento boxes

All things for Sale – bento section

Dino Direct – free world wide shipping!
For Micom Fuzzy Logic Rice Cookers, I have found two suppliers for the UK
Yum Asia – Zojirushi Stockist
Cook Japan – Own Branded and Sanyo Cookers