Can one be a moral nihilist and hold teleological beliefs at the same time? More specifically, can I be a moral nihilist, whilst also being a utilitarian? And further, is teleology a branch of moral nihilism?
Welcome, Matthew. I think some clarification of concepts would help answer your questions.
Moral nihilism is the view that nothing is valuable, there are no binding moral norms, and nothing is worth doing.
Utilitarianism by contrast does posit binding moral norms. It holds that one should promote the greatest happiness of the greatest number, or maximise welfare, or minimise pain and suffering. It invites all sorts of philosophical criticisms but nobody denies, so far as I know, that it posits moral obligations, that it supposes happiness or welfare or the minimisation of pain and suffering are valuable, and that the actions entailed are worth doing. There is complete divergence from moral nihilism on this point.
Teleology assumes the existence of goals. These may be intrinsic natural goals, such as an acorn's non-accidentally growing into an oak tree, which can be seen (e.g. by Aristotle) as the intrinsic natural goal of the acorn's development. Utilitarianism is teleological in a different sense. It does not find an intrinsic natural goal in morality but it sets morality the goal of promoting the greatest happiness of the greatest number, or of maximising welfare, or of minimising pain and suffering.
So to answer your questions :
You can't be a moral nihilist at the time as being a utilitarian. Utilitarianism recognises that certain things are valuable (maximising welfare) and that we have a binding moral norm to promote these things, and promoting them is worth doing.
You could be a moral nihilist and accept certain teleological beliefs. You might believe that that nothing is valuable, there are no binding moral norms, and nothing is worth doing; and yet still recognise the existence of intrinsic natural goals as in the acorn example, which commits you to no moral beliefs at all.
Teleology is not a branch of moral nihilism. To believe in natural teleology of the Aristotelian acorn variety is logically unrelated to moral nihilism. And to accept moral teleology of the utilitarian kind is diametrically opposed to moral nihilism since it posits, what moral nihilism denies, namely (to repeat from 1.) that certain things are valuable (maximising welfare) and that we have a binding moral norm to promote these things, and promoting them is worth doing.