To add a bit more detail, the walkway from the Hilton to T4 is covered, but it is also long, allow 5 - 10 minutes for that part, and it is ever so slightly uphill. To access the walkway, turn left out of the Hilton main entrance and walk along the pavement/sidewalk until you see it.

You can take either the Elizabeth Line or the Underground from T4 to T2/3 for free, however in both cases you still need to pass through faregates. You can get a free transfer ticket for the Elizabeth Line at the station (the gates are sometimes open, but you should get the ticket anyway), or for either the Elizabeth Line or the Underground you can tap your contactless credit card/device to enter and exit (you won't be charged). If you are transferring at an off peak time be sure to check the schedules, either service may be suspended or reduced in frequency.

(I believe the lifts closest to the terminal entrance from the Hilton walkway go to the Elizabeth Line, you need to cross the terminal for the Underground.)

Once you get to T2/3, follow the signs to T3. You will go up an escalator or lift to street level, where you will end up outdoors at the plaza in front of the terminal. The joint VS/DL check-in area is to your left, This is one level down from the Upper Class wing, but there are UC check-in desks on this lower level. Once you check-in, the agent can point you to the lift to the UC security channel.