So at the very least you're looking at London to Toronto via Queen's Park, and then Toronto to Los Angeles via Union Station.
To get back by a different route you could go LA to San Francisco at Mariposa Station, San Francisco to Toronto via Church Station, then repeat yourself slightly by using Queen's Park again to get from Toronto to London. If you wanted to avoid the repeat you could go Toronto to Tyne and Wear Metro(!)'s University Station, and then from Tyne and Wear to London via Monument Station.
The major difficulty appears to be connections from London to anyplace else because the large majority of London's tube stations have idiosyncratic names.
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To get back by a different route you could go LA to San Francisco at Mariposa Station, San Francisco to Toronto via Church Station, then repeat yourself slightly by using Queen's Park again to get from Toronto to London. If you wanted to avoid the repeat you could go Toronto to Tyne and Wear Metro(!)'s University Station, and then from Tyne and Wear to London via Monument Station.
The major difficulty appears to be connections from London to anyplace else because the large majority of London's tube stations have idiosyncratic names.