You can check this yourself using ITA Matrix and comparing the taxes/fees/charges for a one way J ticket to a one way F ticket.
Both list identical charges, so there is no additional cash component required:
USDA APHIS Fee (XA) £2.90
US Immigration Fee (XY) £4.10
US Customs Fee (YC) £3.20
United Kingdom Air Passengers Duty (GB) £138.00
United Kingdom Passenger Service Charge (UB) £44.41
(YQ) £159.50
US International Arrival Tax (US) £10.20
WT->WTP or WTP->CW generally does have additional APD (for WT->WTP) and YQ: