In one store (Midcounties co-op, green) contactless worked fine, but with a transaction - refund - transaction in the feed.
In a second store (The co-operative group, blue) contactless was declined but chip and pin worked and appeared as a single transaction in the feed.
I’ve used the Midcounties co-op store before and not had the 3 part transaction as seen above.
I’ve not had contactless issues using Monzo with Coop food stores in London but there could be differences across the country. I’m not sure that the Coop still operate as separate regional businesses though.
Yes there are separate regional businesses including the main Co-operative society there are some fairly large smaller ones like the East of England Co-operative Society in Essex and Suffolk area
My experience with cards in co-op stores is that contactless can very occasionally be temperamental (even with a Co-op Bank card!). I was told on one occasion that it was because my card had been next to my mobile phone for too long! All is fine 99% of the time and it shouldn’t make much difference between most societies because they often have commonly sourced till systems (eg Group aka blue and Midcounties). If the problem occurs again, and you are a member, then call customer services and have a polite moan (you are a part owner of the business after all).
BTW; there are three big regional societies - Central England (Birmingham to Essex), Midcounties (West Mids to Wilts) and Scotmid. Co-op Group if found everywhere. I’ll stop there, now you are all asleep.
Apart from Central England and Midcounties the East of England is just as big in membership if not area, but Essex also has other tiny Co-op’s too such as the Chelmsford Star. The whole Co-operative movement is sadly fragmented and while many societies recognise other society membership cards and give discounts to students with an NHS Extra card, some like East of England stubornly refuse to recognise the main national membership card or those of neighbouring Co-operatives and refuse to give a student discount.
It’s funny because while not damaging, putting my Android phone next to a contactless card for an extended period of time will send the EMV transaction counter in to the thousands. 
It’s more the magstripe that fails in certain situations I find. Especially rewritable ones used as keys in hotels, etc.
With my Curve Black card they sent a free leather wallet that in theory protects the chip and strip so you can’t read the contactless data thru it and it resistant to accidental magnetism by keys etc (though not resistant to deliberate strong magnetization). The screening of the contactless works (as I tested it with a reader) but not sure about the strip as did not want to risk trying to magnetise it
Yep, there are 500+ retail co-ops in the UK and though they do cooperate, they can also be fiercely independent too.