You can say a lot about Barcelona but not that the weather's grim and damp in January compared to London...
Actually, it is mainly. Yes, warmer than London but it certainly isn't all sunshine and dry at that time of the year. It's a city pretty much geared up to the warmer seasons.
P.S. Maybe I've been lucky in the years I've attended London, but I counted 27 days and nights over last 10 years, of which I've seen rain on two. Most times the weather has been dry and settled, mainly sunny, but yes a bit cold too.
Average annual rainfall:
London 585mm with 107 wet days
Barcelona 640mm with 55 wet days
Madrid 465 mm with 63 wet days
Paris 650mm with 110 wet days
Amsterdam 760mm with 200 wet days
Yet, even so, it doesn't rain that much in London. According to the Met Office Climate data, over the 30 year period, there were 106.5 days of rainfall per year on average (which counts as a day in which 1mm of rainfall or over fell). This means that there was rainfall on 29 per cent of days per year and on average it didn’t rain 71 per cent of days per year. Average rainfall is 557.4mm with 1410 "sunshine hours."
There are more rainy days in Miami (at 135) and Orlando, Florida (117) than there are in London. New York City clocks in at 122 days and 1,268mm of rain. Washington DC, Rio de Janeiro, Sydney, and Mexico City all have more rainy days on average in any given year than London.
In fact, of all Northen European capitals including Brussles, Oslo, Copenhagen etc. London is the driest and has least rainy days per year. Stockholm is about as dry but has far more wet days. The foulest place in Europe is Bergen, Norway which is wettest by a huge margin in Europe and has rain/precipitation on nearly 2/3 days.
P.P.S. Admittedly, after a series of hot dry summers and regular droughts, the UK in 2023 is in the middle of its vilest, worst summer in decades. After a good June, the 'Omega' weather pattern has been fixed in place over the Atlantic for the last 6 weeks now, meaning New England and Nova Scotia on the east of North America, Iceland, The UK and most of Northern Europe are having little sunshine and lots of grey skies and rain, here it's 2-3 times more than usual, in Northern Ireland and NW Egland the wettest July ever. We are having the kind of summer that if you experienced just once in your younger years would make you determined later in life to have an annual holiday in Spain, Greece etc. and never consider one in the UK.