is there any game related to it?
Yes, used in a children's singing game
London Bridge Is Falling Down
The rhyme is often used in a children's singing game, which exists in a wide variety of forms, with additional verses. 
where does the song london bridge is falling down come from Similar rhymes can be found across Europe, pre-dating the records in England; it is possible that the rhyme was acquired from one of these sources what is it all about? It deals with the depredations of London Bridge and attempts, realistic or fanciful, to repair it how did it originate? Similar rhymes can be found across Europe, "Knippelsbro Går Op og Ned" from Denmark, "Die Magdeburger Brück" from Germany, etc. It is possible that the rhyme was acquired from one of these sources what does it mean really? The meaning of the rhyme is not certain
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London Bridge Is Falling Down [SEP] The game
The rhyme is often used in a children's singing game, which exists in a wide variety of forms, with additional verses. Most versions are similar to the actions used in the rhyme "Oranges and Lemons". The most common is that two players hold hands and make an arch with their arms while the others pass through in single file. The "arch" is then lowered at the song's end to "catch" a player. In the United States, it is common for two teams of those that have been caught to engage in a tug of war. In England until the nineteenth century the song may have been accompanied by a circle dance, but arch games are known to have been common across late medieval Europe.