traveling with bike (by train, Bristol->Reading->Gatwick)

I want to travel by train, from Bristol to Gatwick (one change at Reading), are there any problems with bikes? (i know i need to pay ~10GBP for oversized bag), but, should i be worried about available space ? (everything depends on discretion of staff)
train starts 8am Saturday (and about 9.40am from Reading)
train starts 8am Saturday (and about 9.40am from Reading)
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