I would suggest an Edwardian i.e. square one over a Victorian as the amount of usable space you lose from the rounded walls is silly. You are likely going to want something to store the children's stuff in, so drawers units etc - maximise the wall space.
We also had the wall that faces the neighbours built to full height rather than just a view of a fence 3 feet away. It also meant we could put full height bookcases along that wall for storage.
Now is probably a good time to buy as they will be scaling down from the peak and looking for work to keeps the crews busy in the winter period. Ask around locally for recommendations. We used a local firm with a good reputation rather than one of the nationals. We just got a straight price without the usual inflated BS price followed by the gazillion "discounts" palaver.
Conservatory firm should check on any planning and building regs issues. Its very area dependent - for us, nothing special was needed as the increase in floor area we were gaining was below the threshold of needing planning permission.
Underfloor heating is good for maintaining a temperature, but we find it doesn't really warm the conservatory during the winter. We tend to blast it for 10 minutes with an electric fan first. You may not need that if you go with a solid roof type and also depends where you live.
If you don't have an external mains socket, get one done when you do the conservatory. Its so useful when doing power tool jobs outside, no more trailing cables through the window.