How to politely inform potential PhD supervisor that I must wait before responding to an offer? (Two-body problem) I am a physics student in the UK.

A university (university A) has made me a PhD offer. The email from admissions was very clear that there is a 30-day time limit on this offer. As it happens I am likely to accept the offer, but do wish to take advantage of these 30 days. This is because my partner has also applied to university A, and their application is going well, we think it probable that we may both get offers from this university. However, there is a second university (university B) who has already offered my partner a place and will get back to me in 24 days. 

In fact, we will both accept the first of A or B to make us both an offer. Perhaps this is not the best reason to chose a PhD place, but that is beyond the scope of my question. 

The professor who would be my supervisor at university A has emailed me to encourage me to accept A's offer sooner. Clearly, I wish to remain on good terms with this professor, in case I do end up at university A. I'm not sure that telling A about the two-body problem is a good idea. How can I politely tell A that I will not accept yet? 

Edit; I did inform A of my reasons for hesitating. Not only were they very sympathetic, they did their best to get my partner interviewed sooner, and helped them find an intern-ship with the relevant people.
In my opinion, since you are already accepted, there is no harm in telling about the two-body problem; if you end up going to University B, Prof@A.edu would likely feel better about that reason than some negative impression of them personally or of University A.

You are well within your rights to take the full 30 days to decide. Certainly not all professors are equally versed or sympathetic towards the two-body problem, but most have experienced it, if not personally, through their colleagues, friends, etc.

Ultimately I think it is your choice whether or not to explain your reasoning, but I would personally lose a bit of respect towards your Prof@A.edu if they were unable to accept you stating simply "My offer gives me 30 days to respond, and I would like to take that entire time to ensure I make the right decision for me. I will inform you immediately of my decision once it is made."