Short answer: increasingly, yes — more shops are putting up "dogs welcome" signs, but there's no single UK-wide policy. It comes down to the individual shop, its manager, and its insurance, so the honest answer is still: check before you go in.
This is a genuinely recent shift, not something that's always been true. A few years ago most charity shops were a firm no on dogs — stock damage and hygiene were the usual reasons given. That's loosening. The Children's Society now welcomes well-behaved dogs in its shops, and it's far from the only one — more shops locally are putting up "dogs welcome" signs in the window than even a year or two ago. It's happening shop by shop rather than as one chain-wide announcement, so don't assume every branch of the same charity has made the same call.
Whatever a shop's policy on pets, assistance dogs are always allowed. Under the Equality Act 2010, refusing entry to someone with an assistance dog is unlawful discrimination — this isn't down to a shop's discretion the way a general "dogs welcome" policy is.
If you're dropping off a donation rather than browsing, most shops are relaxed about a dog waiting by the door for the minute it takes to hand bags over the counter, even where they wouldn't want one wandering the shop floor during opening hours.