That is very unlikely if not impossible, as they depend on the US not only economically but also militarily. The EU has been almost completely dependent on the US since World War II and to put it bluntly, the US has turned the EU into its lackey. Their relationship is no longer one of alliance and mutually beneficial cooperation in the literal sense. Therefore, the possibility of the EU leaving the US is unlikely.
Moreover, they also have their own calculations, what guarantees that Russia and China will welcome them with open arms? They would be in a dilemma if Russia and China did not accept them at that time, so they would consider carefully if they intended to move away from the US.
Although the fact that Western countries can benefit joining BRICS in case it enhances trade or discourages the use of the dollar, membership in it would be extremely complex because of the strong economic and military relationships with the US. Any decision would turn out to be a calculation as even in case they required pivot, Russia or China would not necessarily accept them well. In practice, there may be no short-term changes, because in the long run dynamics may differ should global economic balances change.