Oasis were The Rain.
The Velvet Underground were The Primitives, then The Warlocks, then The Falling Spikes.
Somebody already said Queen were Smile, but before that they were 1984.
Nick Cave & The Bad Seeds were originally The Boys Next Door, and then had initial success as The Birthday Party, but I think NC&TBS and TBP might actually be seen as two different bands, rather than the same band under a different name.
I believe Sex Pistols were originally The Swankers, hahaha.
Pink Floyd went through numerous names. At first they were Sigma Six, then The Meggadeaths, The Screamin' Abdabs, The Abdabs, Leonard's Lodgers and The Spectrum Five. They eventually settled on The Tea Set until they came across another band with the same name, so they became The Pink Floyd, dropping the "The" after their first album.
Status Quo are another band that initially had a "The" prefix but dropped it after some early success, but before that they had a few single releases that went nowhere as The Spectres, some of The Spectres' output can be found on certain Mod compilations, I believe. They also went through the names Traffic (not to be confused with Steve Winwood's Traffic) and Traffic Jam, before The Status Quo.