Solution 1:

You have an architectural problem, you are abstracting the different types of streams in a bad spot, that cannot easily be untangled. The mock interface is too tightly entwined with the code under test. You can hack around this, but you are better off stopping and thinking through exactly what you want to test.

Sort of side note:

StreamImpl is a bad idea.

StreamImpl& operator<<(T &&out) noexcept;

is non-standard and part of how you got into this problem. If you use the standard

std::ostream & operator<<(std::ostream & strm,  T &&out) noexcept;

and turn StreamImpl into a child of std::ostream you can pass in a re-write of MockStreamImpl that is ALSO a child of std::ostream and completely hide the specialization of the stream inside the stream rather than the function writing to the stream.

Side note to the side note: Templating operator<< isn't a very good idea either. Most types don't have similar-enough rules to be printable by a generic function. This leads to goofy output or confused error messages when the << template tries to invoke itself because it's still the best match.

What went wrong:

If we follow the includes and perform the substitutions we can see what the compiler sees:

#include "MockStreamImpl.hpp"

namespace fixture {
using ::mock::MockStreamImpl;
using ::utils::Dispatcher;
using ::testing::Test;

class UtStream : public Test {
   // Class for setting up mocks and common test data which is used as fixture in tests
   
    StreamImpl kDefaultStream_{};
}
}  // namespace fixture

leads to

#include "StreamImpl.hpp"
namespace mock {
using ::StreamImpl;
class MockStreamImpl {
   MOCK_METHOD(...); // Gmock methods
};
}  // namespace mock
namespace fixture {
using ::mock::MockStreamImpl;
using ::utils::Dispatcher;
using ::testing::Test;

class UtStream : public Test {
   // Class for setting up mocks and common test data which is used as fixture in tests
   
    StreamImpl kDefaultStream_{};
}
}  // namespace fixture

and then to

class StreamImpl {
...
    template <typename Type>
    StreamImpl &operator<<(T &&out) noexcept;
...
};

#include "StreamTemplateImpl.cpp"

namespace mock {
using ::StreamImpl;
class MockStreamImpl {
   MOCK_METHOD(...); // Gmock methods
};
}  // namespace mock
namespace fixture {
using ::mock::MockStreamImpl;
using ::utils::Dispatcher;
using ::testing::Test;

class UtStream : public Test {
   // Class for setting up mocks and common test data which is used as fixture in tests
   
    StreamImpl kDefaultStream_{};
}
}  // namespace fixture

and then, yawn, to

class StreamImpl {
...
    template <typename Type>
    StreamImpl &operator<<(T &&out) noexcept;
...
};

// Templates implementation needed for test purposes
#include "StreamImpl.hpp"
#include "MockStreamImpl.hpp"

using ::utils::Dispatcher;
using ::mock::MockStreamImpl; // Error here

template <typename Type>
StreamImpl &operator<<(T &&out) noexcept
{
    return Dispatcher<MockStreamImpl>::Instance().Get()->OutOperator(out);
}
namespace mock {
using ::StreamImpl;
class MockStreamImpl {
   MOCK_METHOD(...); // Gmock methods
};
}  // namespace mock
namespace fixture {
using ::mock::MockStreamImpl;
using ::utils::Dispatcher;
using ::testing::Test;

class UtStream : public Test {
   // Class for setting up mocks and common test data which is used as fixture in tests
   
    StreamImpl kDefaultStream_{};
}
}  // namespace fixture

and since StreamImpl.hpp and MockStreamImpl.hpp are protected by include guards we get

class StreamImpl {
...
    template <typename Type>
    StreamImpl &operator<<(T &&out) noexcept;
...
};

// Templates implementation needed for test purposes

using ::utils::Dispatcher;
using ::mock::MockStreamImpl; // Error here

template <typename Type>
StreamImpl &operator<<(T &&out) noexcept
{
    return Dispatcher<MockStreamImpl>::Instance().Get()->OutOperator(out);
}
namespace mock { //because mock isn't found by the compiler until here
using ::StreamImpl;
class MockStreamImpl {
   MOCK_METHOD(...); // Gmock methods
};
}  // namespace mock
namespace fixture {
using ::mock::MockStreamImpl;
using ::utils::Dispatcher;
using ::testing::Test;

class UtStream : public Test {
   // Class for setting up mocks and common test data which is used as fixture in tests
   
    StreamImpl kDefaultStream_{};
}
}  // namespace fixture

where we can see that MockStreamImpl is required before it is defined.