My fiancé lives in Dublin, and I spent 2 weeks in June/July traveling through Ireland, London, Paris, and Edinburgh. Honestly, unless you go in uniform, you’re probably not going to scream “cop.” I saw so much FDE and MOLLE on so many obviously non-LE/Mil/gun-toters, culminating with a junior high girl toting a Maxpedition pack on the ferry from Caen to Portsmouth. Personally, I’ve been really happy with North Face stuff for affordable quality gray man but still useful stuff.
Expect temperatures in the mid-40s to mid-60s, and intermediate soft rain, but rapidly changing weather conditions. I like a water-resistant outer layer, and sweaters/sweatshirt underneath if needed (I’m from AZ, so I totally get no cold tolerance). A light jacket should be good...you don’t need heavy-duty stuff, and you won’t get into really cold temperatures.
Go to Galway. One of the coolest towns on this planet. The Quays does an amazing Guinness beef stew. The Churchill War Rooms in London is a can’t miss. If you can pry open the display case with Winston’s Webley, I fall dibs. Let me know if you want other tips for Dublin or London.