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1. We're keeping it quiet from everyone else in work so our interactions with each other around other people are just weird.
It can be very difficult to maintain a relationship with someone you work with. I don't know what the culture is like in the UK, but in the US you can be putting yourself at risk if the relationship is not disclosed to HR or management. I would seriously consider disclosing the relationship ASAP, on a confidential basis.
That said, as long as you're trying to hide a relationship from people, your interactions will always be strange. You're expected to be one thing here and another there, which isn't natural. You'd be more comfortable if other people at work knew about it, probably. The first step is disclosing it to HR or management and determining what the policy is on workplace relationships.
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3. We only really see each other maybe once a week outside work, it seems like she prioritises her friends over me.
Totally normal for being one month into things, like Snarg mentioned.
Get the workplace thing taken care of, unless there is some reason not to.