How can I not look like an American tourist in Edinburgh in January?

I'm a <50 yr old woman from California and I pretty much don't know how to dress for cold and rain nor uneven pavement. I have to visit Edinburgh in January and I would like to spend the weekend walking Old Town and the Royal Mile, seeing the sights (silly I know, but you visit Fisherman's Wharf in San Francisco after all) and popping into a pub or casual ethnic restaurant when I get hungry. I had planned to wear a wool coat, scarf/gloves, jeans and wellington boots. Will I get hled to buy cheap souvenirs? Will the jeans deny entry into a cheap eatery and force me to subsist on the snack bars I've packed into my bag? I know trainers are forbidden even in pubs- what about the wellies? Thank you.