Assuming the child is dressed properly and it isn't extremely cold (I'm thinking high 50ºF or 10ºC with wind). Their hands and face will be exposed and they may be sweaty from running around, but does it impact immunity or health generally to be outside in the cold while young (toddler age, 1-3 or so).
Pediatrician Lynn Wegner says, in this instance, the science seems to be on his side.
"A winter virus is not caused by going out in the cold air," says Wegner. "It's viral transmission." In other words, colds are caught by coming into physical contact with someone who already has a cold virus.
In general no.
A recent study by Eccles is one of the few to suggest a link. Ninety college students dunked their bare feet in a bowl of chilled water for 20 minutes; a control group kept on their shoes and socks while they sat with their feet in an empty bowl.
Volunteers in the bare-foot group were three times as likely to develop cold symptoms within five days, according to the study published last fall in the journal Family Practice.
Eccles' working theory is that many people carry around traces of rhinoviruses — the bugs responsible for the common cold — in the back of their noses. Exposure to chilly temperatures causes blood vessels in the nose to constrict, cutting down the flow of the immune system's white blood cells. The virus then flourishes, triggering cold symptoms.
Criticism of the study:
But there's a flaw in the Cardiff study, says the University of Virginia's Dr. Ronald Turner: The researchers didn't check to see if a virus was ever present. "They measured symptoms," says Turner, a pediatrician who investigates the effectiveness of cold remedies, like echinacea. "They didn't do any virology, so that study has nothing to do with becoming infected," he says.
A 1968 New England Journal of Medicine study remains unchallenged as the definitive word on cold and the common cold, says Turner.
Volunteers were first infected with a cold virus, then chilled and checked for symptoms. The study showed that dampness and cool temperatures didn't increase the chance or severity of a cold.
However, I sometimes wonder if heavy breathing in cold air can irritate the throat and make it more vulnerable to opportunistic viral / throat infections. I don't have any citation to support that, and certainly I wouldn't prevent my child from playing outside all the time just because they might get sick .. so take it for what you will.
Good common sense moderation is perhaps the best advice here -- kids want to play, so let them, but don't stay outside in the cold too long.
When I read the title I thought you were talking about temperatures below freezing, but even then my answer would've been the same. I believe that going outside as much as possible, even when it is cold, improves immunity and reduces susceptibility to the colds, it is also important for mental health and overall development. Some of benefits of spending time outside (improves distant vision, helps to stay fit, reduces ADHD symptoms, etc) are listed here with references to the studies. Honestly, for me 50F is very comfortable temperature and I would not even think twice about going outside with my son. If you are afraid about hands and head exposure, you can put a cap or light hat and gloves on.
It is always important for the child (and adult) to be comfortable: not cold, not sweating. So it means hat, which covers the ears, mittens and warm snow boots when temperatures are below freezing, appropriate layers which can be taken off when warmer. Personally I believe that it is very important to make sure that feet are not cold. Then you can stay outside for a long time, getting a lot of benefits of outdoors.
I would talk to a doctor because this touches on issues that only someone with access to your child's chart would know. A condition like asthma or being born with a heart defect may change the answer. My overall philosophy is that it's not a good idea to give medical advice sight-unseen over the internet; all anyone can tell you are generalities or what works for their child. Even referencing studies isn't always helpful because your child could fit into some kind of out-lier case not covered by the study.
I don't consider 50*F cold at all, and frequently played outside in that kind of weather as long as I was dressed comfortably. That said, I was used to far colder weather, and made sure to wear a scarf over my mouth because I am asthmatic.