Synonym for "godchild" without religious connotation

In English is there a synonym for "godchild" (or the gender-specific versions) without a religious connotation but without necessarily having other specific connotations?

All the words I'm coming up with mean (or at least imply) that the child is living with me, or that I'm acting in loco parentis in some way. I don't want that.

• "orphan", "foster child", "adopted child": these all mean that the child has lost his or her natural parents and is living with a different family or in an orphanage.
• "protégé": this is closer, but to me it still implies that the child lives with the mentor rather than the parents.

What I am looking for is something that describes a similar relationship to "godchild" (i.e. I care about the child's well-being, but he lives with his parents, not with me) without actually having the religious connotation or implying anything about the child.

(As some commenters mention, "godchild" is sometimes used outside of a religious context these days, but it still carries that connotation; it can't really be separated from its origin.)

Both my children, and my four nieces and nephews, all have 2-3 godparents each.

All except one of the godparents are also aunt/uncle to the protectee, if it were felt necessary to avoid the word "god". But despite the fact that all bar none of the godparents are staunch atheists, I don't recall anyone ever objecting to the standard terms godson, goddaughter, godparent.

So far as we're concerned, a godparent is primarily someone charged with taking "greater-than-might-otherwise-be-expected" responsibility for the child's development of a "moral compass". There's often an implication that the godparent is first choice for "adoptive parent", if the biological parents meet with disaster - but that's got no legal significance in the UK, so it's really just a symbolic role.

As @John Y says, for many today the god in godparent has no more religious significance than the x in xmas, or the christ in for chrissake. So I would cite the fact that we don't bother to popularise an alternative "secular" term as evidence that most people simply don't feel a need for it.

In UK in the past, the child would have been called a ward and the adult a guardian, but such situations don't really arise any longer.

"Charge" is another term used for a child under the care of a non-parental adult:

From The Macmillan Dictionary:

[countable] formal

someone that you are responsible for and take care of

Ex. "She didn't like it when her young charges started crying."

The term godchild is getting to be about as inseparable from God as Christmas is from Christ. In other words, godchild is practically devoid of religious connotation when used among secular company.

If you absolutely must avoid the term, the most common way to express what you are after is to use a phrase such as "like a son to me" or "as if she were my own".

Generally such people (at least in parts of North America I'm familar with) are referred to as "uncles" or "aunts". There are meanings of these words that imply a specific blood relationship (siblings of a parent), but that isn't nessecarily the case. It often refers to someone who has roughly the same kind of relationship to the child as a blood uncle or aunt would: IOW they have a close relationship with a parent, care about the child, and might even watch him or her for a while to help the parents out, but are not ultimately responsible for them.

For instance, when I was very young, I had an "Aunt Judy", who was simply the woman who happened to be my mom's roomate in college.

My dad's best friends were all referred to as "Uncle {insert-first-name-here}".

I recently read a piece of fiction written by a Canadian (I'll try to look up the specifics when I get home) where all the women in an extended family were referred to as "aunt" or "auntie", and the men "uncle", regardless of the actual kin relation. So I know this isn't confined to the USA.