The CAA tell me there is no restriction on the carriage of a firearm or a shotgun in a private aircraft in the UK. You have to observe the same rules for transport of the weapon with regard to security the same as for transport in your car. It is the transport if ammunition that is of more note as that comes under the transportation of dangerous goods clause.
In the event of a crash and fire I would not want to add to the problem of a fire with live ammunition!
I regularily fly to shoots and carry my shotgun and occassionally rifle. I have landed at big airports and disembarked with weapons without problem. I never carry ammunition, prefering to save the weight and get it locally or get someone on the shoot to buy it for me.
I have also taken my shotgun to Spain on a commercial flight on a number of occassions without problem. This just requires the correct paperwork to be in place and the weapon to be secured correctly.