The answer to this question depends on how long they have persevered in settling at boarding school and hence how long they have been feeling homesick and how their feelings are being expressed. Some children can for example experience a deterioration in their health, for example if they stop eating through homesickness, or become depressed, excluding themselves from mixing with their peers and making friends. In these cases the school medical support team will undoubtedly become involved in offering your child support and they will be in regular communication with you to ensure you are fully aware of the situation if they feel there is cause for concern.
Keep talking to the school staff team about your concerns and never let on to your child that you are worried, rather encourage them in a positive way to become involved in school life.
Never promise your child at any time that they can leave school if they are still unhappy by a certain date as this will almost certainly lead to their determination to remain miserable and remove any motivation they may have had since they know that there is a potential way out.
Some children however are just not cut out for boarding as they are better suited to being home birds. I would advise keeping in regular communications with your child’s Houseparent and if they let you know that they feel that your child is experiencing more distress than in usual then discuss the options with them and agree and action plan to deal with the situation involving support from both the school and the parents. Set some deadlines with the school staff as to expected progress and take their advice as to the best way forward.
Certainly to take your child away from boarding school in the first term would be, in my opinion, unwise and perhaps may be partly as you the parents miss them too. Persevere at least into the spring term. Remember however that you are obliged to give notice of one term if you wish your child to leave and it can be equally as unsettling to change to a new school in the middle of an academic year when friendships have already been made and there is usually ‘gossip’ around why such a move has occurred in the middle of an academic year. Changing therefore at the end of year 1, is perhaps the best option if you feel this is the only option available to you.