Three weeks in UK, based out of London

Consider @Malta. Even in winter you can find very sunny days and get some nice time near the beach or sailing around the islands. 

I've seen so many tours to Paris from London. I love Paris. It is fairly safe, unlike Belgium and some of those others. Google London to Paris tours. They take you on the train, which is a great experience. We stayed in the Latin Quarter last time, but next July we will be staying on the Champs Ellysee.

Amsterdam: if you intend to visit Anne Frank House (and it’s not to be missed) you must book online ahead of time, do it ASAP. Tickets are timed and limited in number, and if you fail to book you will be in an extremely long queue in the afternoon. The Rijksmuseum Amsterdam is fantastic, and there is an App with several itineraries to help you find your way around the museum - search in the App Store for Rijksmuseum. The A’DAM Toren offers amazing views of the city and is reached by taking the free ferry from behind the Amsterdam Centraal Railway Station. For an easy day trip out of the city I would recommend catching a train to Haarlem, which is like a smaller, quieter Amsterdam still with lots to see. 
Bruges: lots to see in this beautiful small city, and all within easy walking distances within the historic centre. The Belfry of Bruges has many steps but offers great views. Belgium is famous for beer and chocolate, and there is a massive choice of places to enjoy either! 
Paris: the Louvre Museum is amazing, but I love the Musée d'Orsay, which houses the largest collection of Impressionist art in the world within a fabulous former railway station building. For a day trip from Paris I would recommend Giverny, Monet’s former home, or Château de Vaux-le-Vicomte. Either are totally doable from Paris using public transport, no need to pay for an expensive tour. An unusual place to visit is the Père Lachaise Cemetery, a garden cemetery and the resting place of an amazing array of famous people. It’s also the site of the Mur des fédérés, where the Paris commune was finally brought to a tragic end. There is a massive amount to see in Paris, best to just get a guide book and tick off the main sights :) Leave time to just walk the streets and absorb the atmosphere, the Marais District is a great place to do this. Eating out: I would recommend Le Bouillon Chartier as not to be missed. Not possible to book a table and you will probably queue to get in, but it will be worth the effort.
Finally - not sure if you’ve considered it, but Edinburgh is also a great city for a long weekend. 
Enjoy your trip :)

Hi Meredith, All very good options, you can even combine some of them if you take a long weekend (4 days). For instance, Antwerp and Paris are only 2 hours apart by train. Same goes for a lot of cities in Belgium and Amsterdam, only a couple of hours apart. As said by others, the weather might not be great, but there are so many indoor activities in the cities as well. In Amsterdam you can find amazing museums like Rijksmuseum Amsterdam or the Anne Frank House and don't forget the rather unique nightlife.