DLR from London Tower to Cutty Sark

Go from Tower Gateway and change enoute. Oyster works throughout the DLR but validators are often standalone or attached to walls etc rather than being on gates. Don't forget to use them.

The London transit planner is a useful tool: https://tfl.gov.uk/plan-a-journey/
Assumed travel date and time Tower Hill station to Cutty Sark:

Tower Hill Underground Station To: Greenwich Town Centre / Cutty
Sark Leaving: Friday 25th May, 12:00

Tower Hill Underground Station at 11:55via District line or Circle
line to Monument District line or Circle line to Monument

Journey time: 1 min Step 2 DepartMonument Underground Station at
11:56via Get off at Bank Get off at Bank

Journey time: 4 min Step 3 DepartBank DLR Station at 12:00via DLR to
Cutty Sark (for Maritime Greenwich) DLR Station DLR to Cutty Sark
(for Maritime Greenwich) DLR Station

Journey time: 18 min View stops Step 4 DepartCutty Sark (for Maritime
Greenwich) DLR Station at 12:18via Get off at Greenwich Town Cen/
Cutty Sark Get off at Greenwich Town Cen/ Cutty Sark

Journey time: 7 min Arrive atGreenwich Town Cen/ Cutty Sark at 12:25
Greenwich Town Cen/ Cutty Sark

An alterntive route if you would like to walk under the River Thames (via the Greenwich foot tunnel):
https://en.wikipedia.org/wiki/Greenwich_foot_tunnel

Tower Hill Underground Station
To: Island Gardens DLR Station
Leaving: Friday 25th May, 12:00

Tower Hill Underground Station at 11:55via District line or Circle
line to Monument District line or Circle line to Monument

Journey time: 1 min Step 2 DepartMonument Underground Station at
11:56via Get off at Bank Get off at Bank

Journey time: 4 min Step 3 DepartBank DLR Station at 12:00via DLR to
Island Gardens DLR Station DLR to Island Gardens DLR Station

Journey time: 16 min View stops Arrive atIsland Gardens DLR Station
at 12:16

Then walk under Thames to Cutty Sark:

Island Gardens Isle of Dogs, London, United Kingdom

Head southeast toward Manchester Rd/A1206 171 ft

Turn right onto Manchester Rd/A1206 72 ft

Turn left onto Douglas Path 407 ft

Continue straight onto Thames Path (NE Extension) 62 ft

Continue onto Greenwich Foot Tunnel Take the stairs
0.3 mi

Continue onto Greenwich Church St 213 ft

Turn right at College Approach/A206 85 ft Cutty Sark London SE10 9EJ,
United Kingdom

"I may take the City Cruise"--There are also the Thames Clippers that take you to Greenwich. They are fast and take the Oyster card as payment.
Go to the pier right beside the Tower of London, board the Thames Clipper boat (make sure you're boarding the one for Greenwich, not Westminster), tap in with your Oyster card, relax all the way to Greenwich. Since your questions about the DLR have been answered, just thought I'd throw that in. When you get off the boat in Greenwich, tap out with the Oyster card.
To me (warning--personal opinion coming up) this is the easiest and most fun way to get to Greenwich.

Rebecca, I picked City Cruises because it had the open deck to see Tower Bridge and the narration. My understanding is that the Thames Clipper has neither, but is faster. I do like that I can you my Oyster card.

Edgar, this is one of those cases where the TfL webplanner algorithm gives recommendations that are completely stupid if you have any local knowledge. District to DLR at Bank/Monument is a long multi-level journey through a notoriously overcrowded and confusing station complex. Take the DLR from Tower Gateway, get off at Shadwell, and wait for a Bank to Lewisham train there on exactly the same platform. Get off that one at Cutty Sark.

Phillip, thank you! I was going to take the DLR at Tower Gatewsy but wasn’t sure which stop to change. The planner suggested westferry but I will use your recommendation. I appreciate you expertise.