Not really a Target equivalent in the UK, sadly. It needs one!
Dunelm is pretty similar to BB&B. Most of our rugs, curtains, sheets, etc. are all from there. Good quality and selection- it's our go to for anything home-related.
I really like The Range and TK Maxx for home decor and other things. I know not everyone is a TK/J Maxx kind of person, but I've found lots of nice things for the home in there- including a lamp and kitchen supplies. Some TK Maxx stores have a better selection than others.
My husband refuses to shop/have anything from IKEA, so we can't go there

Thinks it's all "cheap tat" and doesn't understand why it's so popular. Don't share his view, but whatever.