You may not use the lanes without the card, as the card contains electronics which tell US Customs that the vehicle and passengers are allowed to use it. See the following from the US Customs & Border Control's FAQ page regarding Global Entry:
Must I obtain a Global Entry card?
No, it is not necessary to obtain a Global Entry card to continue
using Global Entry at airport kiosks. A card is only required for
expedited entry at the SENTRI and NEXUS lanes coming into the United
States.
I'm assuming that you applied for Global Entry as a Mexican national? It doesn't appear that UK nationals can apply for Global Entry:
Can I join Global Entry if I am not a U.S. citizen or U.S. lawful
permanent resident?
Global Entry is also available to citizens of the Netherlands who are
enrolled in FLUX and Korean Smart Entry Service members. Citizens of
Germany, Panama, and Mexico may also apply for Global Entry. Canadian
NEXUS members have Global Entry benefits, but are not eligible to
join.
If so, you should be able to get a card:
Global Entry cards are issued to Global entry members who are U.S.
citizens, U.S. lawful permanent residents and Mexican nationals.
It takes a bit of time for a card to arrive in the mail after approval, but if you're sure you didn't get sent one, you can request one for $15 from within your TTP profile
Though, even after jumping through all those hoops, it probably isn't worth it if you're only crossing once, as you also need to register your vehicle with CPB and have them inspect it prior to using the lane:
Global Entry members who wish to drive their vehicle in the SENTRI
lanes, must register their vehicle with CBP and schedule an
appointment at a SENTRI enrollment center to conduct a 7-point vehicle
inspection.
