People have discussed Covid so will not enter that discussion but NY and UK in winter may have snow / ice and bad travelling weather. The UK is lousy in dealing with snow so be aware, personally I have a 4x4 car with snow tyres so I am ok, (I had to get to work regardless of the weather). I have been in NYC, Cardiff and London in January with snow (no 4x4 car) and not fun as I was working and conferencing.
January in UK is quiet post Christmas and New Year celebrations, most people just hunkering down waiting for spring. Good time of year to be out walking and getting fit for the spring and summer. Personally I would go for May - warmer, lighter nights, spring has sprung and just a better feeling.