Cotswolds in January?

I think this is doable if you are prepared to be rained out. The thing about January in the UK is you could get days that are perfect walking weather (sunny but chilly) or you could get pelting rain. Also keep in mind how short the days are, so you need to ensure you’re finished with your walk by about 4pm when it starts to get dark. Alternatively if you stay in London you have a lot of back up options for poor weather days. Good luck with your planning!

Apart from walking there are bus rides on an every two hours or so frequency to Stratford on Avon, or every two hours to Cheltenham, for a poor weather day.
You could do Blenheim from Chipping Campden- bus to Moreton in Marsh, train to Hanborough then walk in on the estate carriage route- you don't have to go to Oxford for a bus.
Or on a Tuesday or Thursday take a bus ride on the Hedgehog Bus (a Community owned service) to Evesham for a couple of hours there.
Another wet day rail trip would be from Moreton in Marsh to Worcester.

So yes it's chancy weather wise for walking (as said above it could be rain all the time or that special crisp beautiful mid winter sunshine).

As long as you're realistic and flexible I don't see why it couldn't work.

We stayed at the Volunteer Inn in Chipping Campden for six nights, visiting the Cotswolds, Oxford, Blenheim Palace and Stratford Upon Avon. It was in October and not cold or wet.

I would never plan to visit the Cotswolds in the Winter, can't imagine enjoying hiking in that weather.
Still, you could do indoor stuff like Blenheim Palace, Oxford and Stratford Upon Avon (loved the Shakespeare tour of four sites).

I think when you're as far south as Oxfordshire, January isn't so bad as long as you're flexible, like Stuart says. It's not like being out in the wilds of Scotland somewhere.

What I do remember reading on here though, not personal experience, is that some of the popular walking paths can get quite waterlogged in winter. I think mud may be a feature. Welly boots are probably a good idea for The Cotswolds at this time of year, but not very practical to travel with.

When you add up the reasonably high chance of a lot of rain, the almost certainty of a lot of mud, probable grey days, many places being closed, including a lot of pubs for half the week and for me the biggest factor, very short daylight hours I really wouldn’t do it. It is conceivable you could get a week of cold, crisp, bright days but the odds are probably against it. Three young kids might very well hate every minute.

I would stick to more urban areas, much more for the kids to do indoors and plenty of outside options if you do get some fine days.

January is very dark. This year I think we had less than an hour of sunlight for the whole month. The countryside will look very bleak and you’ll need to have walking boots and decent waterproofs for the entire family, not to mention hats, gloves etc. The ground will be too wet and muddy to walk in sneakers even if it’s not raining. It’s really going to be quite a lot of hassle.

I love the Cotswolds.

We don't walk anymore because of health issues, but when I was younger I used to wander all over the northern and western Cotswolds. It is a beautiful and serene place.

That said, honestly January is about the worst time to enjoy walking in the fields. Neither the fields or roads dry out no matter how long since rain because the sun (hah - what's that?) is so weak that nothing evaporates.

The only way you can hope to be successful is fully kitted out in wellies for all, with layers and the top two layers waterproof (not showerproof). Hats, gloves, hand warmers too. There is so much mud that you must prepare for everybody slipping and going down during the hikes. At field gates you should expect fairly deep puddles on and in the mud - livestock and farm vehicles churn up the mud near gates.

Some farmers of particularly hardy breeds of sheep, particularly hill sheep, may keep them out all winter if they can get sufficient feed to them, but between the mud and no food (grass) growing most farmers will have the sheep indoors, so you chance of coming across them is relatively low in January.

Be sure that all walkers have good lights (even if what passes for daylight is nominally from 8:30 to 3:30 or 4, it will sometimes be headlights on all day) and be sure to take a fully charged phone with UK calling. Put What3Words on your phone so if you get lost you can tell people exactly where you are. The emergency number in the UK is 999.

I don't say any of this to scare you or put you off, but I really want you to be prepared.

You'll know when too much is too much for the kiddoes.

British winters tend to be either cold and dry or warm(er) and wet. As a result of global warming, we have had more of the latter recently than we did previously. So, it is now unusual to get several days and nights where the temperature stays below freezing (I said "warmer") and winters without snow in the south of England are becoming quite common. but there can be a lot of rain. The trouble is that if you are planning to hike is that wet weather can be more of a problem - lots of mud, possibly flooded areas in fields and so on.

I'd stay in London or Oxford and aim to walk if the weather allows but accept that it might not.