Is it possible to have an “auto” member variable?

You can, but you have to declare it static and const:

struct Timer {
    static const auto start = 0;
};

A working example in Coliru.

With this limitation, you therefore cannot have start as a non-static member, and cannot have different values in different objects.

If you want different types of start for different objects, better have your class as a template

template<typename T>
struct Timer {
    T start;
};

If you want to deduce the type of T, you can make a factory-like function that does the type deduction.

template<typename T>
Timer<typename std::decay<T>::type> MakeTimer(T&& startVal) {   // Forwards the parameter
   return Timer<typename std::decay<T>::type>{std::forward<T>(startVal)};
}

Live example.