Imamartian, your situation is completely different as the sediment will drop out of suspension in the first butt and fall to the bottom, so little will get transferred down the pipe to the second butt.
Also after rain, your first butt will fill up quickly as I assume from your post that it is connected in the usual way via a short pipe from the diverter, so you don't have the long pipe with loop to create the resistance situation. The 5 metre transfer to the second butt is assisted with the build up of pressure of the water as the first fills. (A 250 litre butt will contain a quarter of a tonne of water when full)
Also there is no mention of a 'loop' in the pipe as proposed by bopcol, where the water drops and is then forced to rise again. This is the issue, as the sediment would settle in the bottom.
Although you do get sediment build up in your diverter (and I imagine a fair bit still makes it through to the first butt which it won't with bopcol's proposal where it will all sit in the loop), you say that your roof is a conservatory, so being glass it won't trap as much debris and dust and will not have particles that wear off as some other roof surfaces may do. In addition, 3 x 4 metres is quite a small roof and would not collect as much debris as a larger area would. bopcol doesn't mention the size or construction of his roof, but it could quite easily produce more sediment than yours.