Effectively use async/await with ASP.NET Web API

I am not very sure whether it will make any difference in performance of my API.

Bear in mind that the primary benefit of asynchronous code on the server side is scalability. It won't magically make your requests run faster. I cover several "should I use async" considerations in my article on async ASP.NET.

I think your use case (calling other APIs) is well-suited for asynchronous code, just bear in mind that "asynchronous" does not mean "faster". The best approach is to first make your UI responsive and asynchronous; this will make your app feel faster even if it's slightly slower.

As far as the code goes, this is not asynchronous:

public Task<BackOfficeResponse<List<Country>>> ReturnAllCountries()
{
  var response = _service.Process<List<Country>>(BackOfficeEndpoint.CountryEndpoint, "returnCountries");
  return Task.FromResult(response);
}

You'd need a truly asynchronous implementation to get the scalability benefits of async:

public async Task<BackOfficeResponse<List<Country>>> ReturnAllCountriesAsync()
{
  return await _service.ProcessAsync<List<Country>>(BackOfficeEndpoint.CountryEndpoint, "returnCountries");
}

Or (if your logic in this method really is just a pass-through):

public Task<BackOfficeResponse<List<Country>>> ReturnAllCountriesAsync()
{
  return _service.ProcessAsync<List<Country>>(BackOfficeEndpoint.CountryEndpoint, "returnCountries");
}

Note that it's easier to work from the "inside out" rather than the "outside in" like this. In other words, don't start with an asynchronous controller action and then force downstream methods to be asynchronous. Instead, identify the naturally asynchronous operations (calling external APIs, database queries, etc), and make those asynchronous at the lowest level first (Service.ProcessAsync). Then let the async trickle up, making your controller actions asynchronous as the last step.

And under no circumstances should you use Task.Run in this scenario.

It is correct, but perhaps not useful.

As there is nothing to wait on – no calls to blocking APIs which could operate asynchronously – then you are setting up structures to track asynchronous operation (which has overhead) but then not making use of that capability.

For example, if the service layer was performing DB operations with Entity Framework which supports asynchronous calls:

public Task<BackOfficeResponse<List<Country>>> ReturnAllCountries()
{
    using (db = myDBContext.Get()) {
      var list = await db.Countries.Where(condition).ToListAsync();

       return list;
    }
}

You would allow the worker thread to do something else while the db was queried (and thus able to process another request).

Await tends to be something that needs to go all the way down: it is very hard to retro-fit into an existing system.