To get the filename from a URL, you can retrieve the HEAD information and check Content Disposition header field. For this task, you can use the TIdHTTP indy component. If the Content Disposition does not have the file name, you can try parsing the URL.

uses
  IdURI,
  IdHttp,
  SysUtils;

function GetRemoteFileName (const URI: string) : string;
var
  LHttp: TIdHTTP;
begin
  LHttp := TIdHTTP.Create(nil);
  try
    LHttp.Head(URI);
    Result:= LHTTP.Response.RawHeaders.Params['Content-Disposition', 'filename'];
    if Result = '' then
     with TIdURI.Create(URI) do
      try
       Result := Document;
      finally
       Free;
      end;
  finally
    LHttp.Free;
  end;
end;