Is it possible to get a PhD position with only a Bachelor's degree?

It's certainly possible -- in fact, in the USA it's more common to start a PhD directly after a Bachelor's than after a Master's. In Europe, having a Master's first is a more typical requirement, but I have known some people to start PhDs straight after their Bachelor's (in physics, in the UK). Having significant research experience and published papers will certainly help your application a lot.

Why not apply for PhD and Master's positions at the same time? That way, you may get accepted to a PhD, in which case great, if not then you will likely have the Master's to fall back on.

I'm starting a PhD program in January. I won't have technically finished my Bachelor's at that point as due to some scheduling issues I'll still have one module left.

If you get the grades and can persuade the supervisor you'll be a good fit for the role, anything is possible.

I'm in the UK.

In Denmark, at least at the University of Copenhagen, they have a 3+5 and 4+4 PhD programs:

In the 3+5 system, you start on an integrated Masters and PhD process straight after graduating with your Bachelors.

It exists at the Aarhus University as well, see section "I'm doing/have a Bachelor's degree". The other Danish universities may have something similar, you can google.