The answer to your question is yes and no depending on how your algorithm is designed. If your algorithm is designed to handle cold start problems, then you don't need to retrain the algorithm. If not, there is need for retraining the algorithm because your system will find it difficult to generate any recommendation for the new user for the fact that it doesn't have the profile of the new user, so there is no way it can compare the user with other users in order to fiind his/her neighbours. Hybridization techniques can help you solve this problem. That is, combining your Item-basedKNN with some item content information(content-based filtering). You could google to see how it is done.
first of, i'm not familiar with the specific library. With SVD and SVD++ the algorithms use user's ratings or purchases in order to fill in the blanks. In this sense these algorithms are not "trained" (training and inference are one step) so there is no concept of retraining. KNN stores information about users' preferences and then at inference they find the closest k neighbors and give a prediction based on that. Even if the user is new to the system (previously unseen) the prediction is made in the same way. By retraining the system you add more information so that better predictions can be made.
BUT as Folasade said the user's vector has to be compared to other users' vectors (in order to find the k nearest ones). If by new user you mean that the user has just signed up (no activity), then you have nothing to compare him with. This is the cold start problem. One way to solve this is by using alternate information about the users like sex, location etc. in order to have some similarity measurement. An other approach suggested in literature is to suggest the most popular items in general.