Originally Posted by Cymro

A lot of those places will be much, much easier to connect to by train rather than by plane, especially when you consider that your only viable OneWorld options would be AirBerlin or BA (so you can only connect through German airports or London).

No European airline really has a point-to-point network - it's all hub and spoke, and they tend only to have one hub. In addition, Avios are redeemed by segment, so you pay 4k + 4k for AMS-LHR-BRU (plus taxes) even though the cities are only a couple of hours apart by road or rail.

So... how can you make the best use of your Avios? Firstly, you don't say where you're based, but if you're located somewhere where AA has a monopoly on feeder traffic, a redemption could be very cost-effective for your first leg. Secondly, you don't have to fly to London to start your trip. So you could easily fly IAH-LHR-BUD and fly back FLR-LHR-IAH, actually earning (a few) more avios on a paid ticket, rather than using them for those connections.

Knowing your specific dates would also help get better value.

So with that in mind, I'd probably end up with London somewhere in the middle of my trip, so maybe something like

BUD - VIE - PRG on ground transportation
Avios PRG - LHR
LHR over a weekend (it's an expensive city for accommodation and tends to get cheaper on weekends)
LHR-BRU or PAR or AMS on Avios, connecting between these cities by train
Then LGW/LHR to FLR/VCE to finish the trip (7,500 avios for this longer flight), with a return flight back to your point of origin.

But, depending on your available time and funds, Non-NonRev's suggestion to prune a little seems like a sensible one.