As you are not changing your first flight, no need to reprice your ticket.
You say you want to change the rest of your itinerary after LHR, your LHR stop becomes a stop over instead of a transfer. If this involves additional taxes and i suppose it does, you would have to pay those. Plus usd 125 change fee as your ticketed points may remain the same, but i think one pays the change fee for changing stop overs into transfers or vice versa. Additionally, airlines may change you a service fee, depending on the airline, for handling these changes.

Actually, the usd 125 fee is questionable: i got away with changing from stop overs to transfers a few times on a RTW ticket without paying the usd 125.
This was not on a OW round the world but on Star and involved Lufthansa both times. I think they wanted me away from their counter as quick as possible and conveniently forgot to charge me for extra taxes and a change fee No service fees either (LH airport desks in SIN and ATH).