Is it legal to build and fly an international aid drone?
Is it legal to build and fly a scale 1m:20cm Boeing C-17 for use in flying from the U.K. [...]

Building an aircraft is likely not regulated (intellectual property laws still apply), it's probably like building a car in your garage, however using an aircraft, specially outside your own property is regulated. Drone use, in UK, is regulated by CAA as for any aircraft, the main reference is:


Air Navigation Order 2016 and Regulations (CAP 393).


Article 23 (Exceptions from application of provisions of the Order for certain classes of aircraft) specifies that some aircraft are excluded from the scope of the order, this includes:


  
  any small unmanned aircraft.
  


So some drones can be used quite freely. Indeed you are always responsible of what happens with your drone. Some articles still apply and in particular art. 94:


Art. 94 - Small unmanned aircraft: Need to fly safely, and can't drop anything that can endanger persons or property. Must maintain direct, unaided visual contact with the aircraft. Unless authorized by ATC, weight is limited to 7 kg when inside A, C, D or E airspace, when close to an airfield, or when flying higher than 400 ft.


This article alone prevents free use in your case.


Source


For the sake of reference, let's look at the other articles:


Art. 91 - Dropping articles for purposes of agriculture etc. and grant of aerial application certificate: This is not allowed without authorization.
Art. 92 - Mooring, tethering, towing, use of cables, etc.: Use of cables being a hazard for other aircraft, some cases are prohibited, e.g. using cables in a controlled airspace, or near an airfield.
Art. 95 - Small unmanned surveillance aircraft: Not applicable.
Art. 241: Endangering safety of any person or property: Don't, even recklessly or negligently!
Art. 239 and 257: Power to prohibit or restrict flying: You may be prohibited to fly if circumstances call for; Art. 265: Offences and penalties: How you'll be fined if you do something not allowed.



For uses not covered by art. 23, CAA says:


  If you want to use a small unmanned aircraft or drone outside of the
  operating limits set out in the Air Navigation Order then you will
  need Permission from the CAA, even if your activity is non-commercial.
  Applications for this type of Permission are considered on a case by
  case basis.



Permissions are granted on a case by case basis, they may be valid for one flight or for a period of up to 12 months.
An operating manual must be prepared and submitted to the CAA, and a designed pilot must be assessed by CAA as having the competences required to fly the missions.


In addition, as mass increases, regulation becomes more stringent:


  Unlike small drones, unmanned aircraft with an operating mass of more
  than 20 kg are subject to the whole of the UK Aviation regulations (as
  listed within the UK Air Navigation Order – ANO), although they may be
  exempted from certain requirements by the CAA.
  
  Unmanned aircraft with a mass of more than 150 kg are also subject to
  additional certification requirements as determined by the European
  Aviation Safety Agency - EASA.


In the UK, "large" drones 


  are not currently permitted to fly in any non-segregated airspace [...]
  without specific permission from the CAA.



Flying from the U.K. to other countries [...]

UK is not part of the Schengen Area, flying to a Schengen Area country will be considered as if flying to a foreign country.

Flying to (or over) any other country requires to follow local laws, which vary, but likely include:


Getting an authorization to fly a drone from the local agency equivalent to CAA. Regulations may be more limiting than in UK.
Getting a specific authorization if the pilot is a foreigner. In some countries this may be not allowed at all.
Getting an authorization to use a camera. Some countries don't welcome flying cameras, some just exclude their use at specific locations.


ICAO Appendix 4 to Annex 2 of the Chicago Convention (Remotely Piloted Aircraft Systems) lists the few agreed principles between ICAO members, including:


  
  An  RPA  shall  not  be  operated  across  the  territory  of  another  State,  without  special  authorization issued by each State
  in which the flight is to operate [...]
  An  RPA  shall  not  be  operated  over  the  high  seas  without  prior  coordination  with  the  appropriate ATS authority.
  



To deliver aid in places that are hard to reach, for example, on top of a large hill where it is hard to get aid workers on foot up there [...]


UPS-backed Rwandan blood deliveries (source)

Why not launching your drone close to the destination, instead of flying from the UK, think about aspects like:


The ability to maintain a radio link with a significant data rate for the camera. This can be difficult without using a satellite.
A drone limited range, even if fuel is used in place of batteries.






More on unmanned aircraft in UK airspace:


Drones are administratively known as UAS (unmanned aircraft system) and SUAS for the small ones.
CAA site
Unmanned Aircraft System Operations in UK Airspace – Guidance (CAP 722)
The Regulatory Framework for RPAs (EASA)