Your assumption that funds sold in GBP trade in GBP is incorrect.
In general funds purchase their constituent stocks in the fund currency which may be different to the subscription currency. Where the subscription currency is different from the fund currency subscriptions are converted into the fund currency before the extra money is used to increase holdings. An ETF, on the other hand, does not take subscriptions directly but by creation (and redemption) of shares. The principle is the same however; monies received from creation of ETF shares are converted into the fund currency and then used to buy stock. This ensures that only one currency transaction is done.
In your specific example the fund currency will be USD so your purchase of the shares (assuming there are no sellers and creation occurs) will be converted from GBP to USD and held in that currency in the fund. The fund then trades entirely in USD to avoid currency risk. When you want to sell your exposure (supposing redemption occurs) enough holdings required to redeem your money are sold to get cash in USD and then converted to GBP before paying you. This means that trading activity where there is no need to convert to GBP (or any other currency) does not incur currency conversion costs. In practice funds will always have some cash (or cash equivalents) on hand to pay out redemptions and will have an idea of the number and size of redemptions each calendar period so will use futures and swaps to mitigate FX risk.
Where the same firm has two funds traded in different currencies with the same objectives it is likely that one is a wrapper for the other such that one simply converts the currency and buys the other currency denominated ETF.
As these are exchange traded funds with a price in GBP the amount you pay for the ETF or gain on selling it is the price given and you will not have to consider currency exchange as that should be done internally as explained above. However, there can be a (temporary) arbitrage opportunity if the price in GBP does not reflect the price in USD and the exchange rate put together.