How big is your garden? 2.5m is the max height allowed within 2m of the boundry. Otherwise Max height can be 4m ( dual pitched roof, 3m for a single pitch ) but no higher than 2.5m at the eaves
A 3m or 4m roof height will give you a lot more headroom, will be a lot cheaper to construct and easier to weatherproof. At a guess a flat roof will be twice the cost of a pitched roof and built by someone with more than basic DIY skills. Dont underestimate the weight of snow even if you are in london it would be a shame if after it’s built London gets the brunt of the bad weather the following year. Use a construction grade timber for the roof ie. C24 ( dry-graded ).
A hardwood cladding would be best for longevity as would a raised floor allowing the air to flow underneath will prevent rotting.