It's dead simple, since you are starting with pounds, just get the pound one.
At the end, you can just take pounds out.
If you went with the USD one, you'd have to "change twice".
Thus, they're simply offering the pound one as a "convenience to you" - so to speak.
(Note that inside their enterprise, they essentially "change on every transaction" in some sense, but since they do so much all at once it is frictionless.)
I'd guess the reason they added a "GBP one" is just because they have many customers there and also they're on that exchange.
(For example, say they suddenly had a pile of Canadian customers, they'd probably offer a CAN denominated one.)
(Or, for example, if you live in Hong Kong the physical-gold accounts (every bank account has one) are certainly denominated in HKD. And you can be sure that in Switzerland there are innumerable CHF denominated gold stores.)
One point, you know how you mention "the gold spot price is in dollars". I mean, you can certainly get a quote on gold (in small or enormous quantities) in all major currencies. I bet you that internally, they often "buy physical gold" using their piles of pounds as well as their dumpsters of dollars. (But none of that matters to you, it's all nominal.)
So your Question Title,
Does trading currency choice in a Gold ETC make a difference when base currency remains same?
It makes a "difference" because - extremely simply - you'll save a few quid by not having to do a exchange at each end.
Your idea that the "base currency is the same" is not totally accurate, you're a bit off track there. (A) As I mention, I bet you a ounce they absolutely buy gold with their pounds; and (B) you say "the gold spot price is dollars" - that's not really the full picture of the situation. Bazillions in bullion is traded in all major currencies all the time; as we speak.