More times than I care to count, I find myself in a situation where I speak to a child in his/her mother's presence and the child does not respond in a way that the mother considers polite. Here's an example:
Me: Hello, Billy, how are you doing today?
Billy: (Ignores me - possibly because he is shy or doesn't want to talk.)
Billy's Mom: Billy, say hello to Mrs. Schwartz and tell her how you are.
Billy's Mom: Look at Mrs. Schwartz when you tell her how you are.
Billy: I'm fine.
Me: I really like the color of your shirt; it looks really good on you.
Billy's Mom: What do you say, Billy?
Billy: I like the color, too!
Billy's Mom: What do you say to Mrs. Schwartz, Billy?
Billy: Thank you.
Billy's Mom: Thank you, Mrs. Schwartz.
Billy: Thank you, Mrs. Schwartz.
This is awkward for everybody. I end up feeling like a heel because Billy gets corrected in front of me. It wasn't my intent to make him feel self-conscious or to make his mom feel inadequate and feel that she has to prove to me that she really does train her children. Next time, I might refrain from saying anything so that Billy and his mom don't have this struggle publically.
I have raised children and am well aware that in spite of the hours of training you put in, children don't always reflect that training. Rather than go through the What do you say? routine, it would be better to make a mental note that more work needs to be done in this area and handle it privately. Pointing out to Billy upon arriving home that his behavior lacked grace and respect is far better than to take what was a friendly gesture and turn it into a character lesson.
You should explain to your children that a corollary to God's commandment to honor your parents is honoring other adults. If there are going to be consequences for failure to obey this, take care of it privately rather than publically. Manners are an important part of life, but encouraging lip-service in a reluctant child in order for the mother to save face is not worth the effort. Do we really want to have our children practice superficial communication?
I love speaking with children and am willing for them to warm-up to me and establish a relationship. It doesn't have to happen immediately. In a world where children are taught not to talk to strangers, it is important for parents to realize that many adults are, in fact, strangers to kids even if they attend your church. You might even want to make a point to introduce your children to people, having practiced with them at home. One of my greatest joys is to see the boys and girls I met when they were little grow into responsible, Christian adults. When these relationships are allowed to develop in a natural way, solid fellowship can result.