If a child has problems with attachment, will this lead to other psychiatric problems as an adult?

Likely. Despite the importance of secure attachment (parent-child relationship), events later in life also play an important role in shaping one's emotional health and capacity for intimate human relationship. The children who had an disorganized/disoriented pattern of attachment (abused children, children of parents with major depression, alchol use and bipolar disorder) are at the highest risk.
Yes. Early attachment difficulties have been shown to be associated with a variety of important adult psychiatric conditions. The child is best served by getting treatment now.
Early intervention. Early problems with attachment are thought to sometimes lead to problems with emotional regulation and stable relationships later on, and/or can be signs of an underlying medical or psychiatric disorder. Getting treatment early can help address the problem before it really takes root.
Yes . It is possible that the child will develop conditions which are common in childhood like adhd or conduct disorder, and through lifetime, other psychiatric disorders like anxiety, depression, personality disorders, substance abuse. The treatment of attachment disorders should include psychotherapy, and psycho-education.