Getting attached to your charge is simply a part of the territory... especially if you've been with him since birth, for a long time, and care for him for the better part of the day (i.e. his waking hours).
Being a nanny is no ordinary job, because you develop a special relationship with your charge over time, and it's completely natural and normal. (It's abnormal, imo, if you don't develop some sort of healthy, attached relationship to them and them to you).
I plan on making it a point to keep in touch with my current charge even after I leave this position. The LO is important to me, I'm important to her, and we wouldn't have it any other way. The parents also agreed that they would love it if I continued to take the LO out on occasion when this position is over, and it will be at my own expense, so everybody wins!
I'm also super attached to the children of a close family friend. I started looking after her oldest child when he was 9 months old and I was in my early teens. He was super attached to me (would cry when I gave him to his parents or relatives), and I loved him soooo much. I treated him as if he were my own child, even though I had the sense back then to know better.
I've looked after her 3 children extensively since birth, and even though I'm not their primary childcare provider anymore (one's in 4th grade, the other in pre-k, and the last in daycare), I still make it a point to visit regularly, spend the night whenever I want, and spend hundreds of dollars on clothes, presents, and outings for them.
I know the above situation is different from your own, but I see no reason as to why a nanny shouldn't develop a strong relationship with her charge, and why she can't, or shouldn't, visit her "ex-charges," if they live a reasonable distance away, the position ended amicably, and she's on good terms with the parents. Taking your ex-charge(s) out for a weekend afternoon would be enjoyable for all parties involved.
It's human nature to love and dote on the LOs you care for, and to want to continue to watch them grow and spend time with them, even after a position is over. Like I said, being a nanny isn't your average position.
Yeah, I want to have a husband, a brood of children... like 10, lol (biological and adopted), and live my own life too, but I'll make it my goal to do it in a way that makes time for all my "other kids" as well.