Yhyjyk

My target is to send from the ERC20Wrapper.sol deployed contract BAT tokens to my hard coded address.

I've created ERC20 Interface:

pragma solidity 0.4.18;

interface ERC20 
 function totalSupply() public view returns (uint supply);
 function balanceOf(address _owner) public view returns (uint balance);
 function transfer(address _to, uint _value) public returns (bool success);
 function transferFrom(address _from, address _to, uint _value) public returns (bool success);
 function approve(address _spender, uint _value) public returns (bool success);
 function allowance(address _owner, address _spender) public view returns (uint remaining);
 function decimals() public view returns(uint digits);
 event Approval(address indexed _owner, address indexed _spender, uint _value);

Then I Added ERC20Wrapper.sol:

pragma solidity ^0.4.18;

import "./ERC20Interface.sol";

contract ERC20Wrapper 
 ERC20 constant internal BAT_TOKEN_ADDRESS = ERC20(0xDb0040451F373949A4Be60dcd7b6B8D6E42658B6);
 address myAddress = address(0xAD53363200C71751FA325ED7bE483722256C3501);

 function BATSend(uint tokenAmount) public payable
 ERC20(BAT_TOKEN_ADDRESS).transfer(myAddress,tokenAmount);
 

Then I instantiate ERC20Wrapper.sol and got its address on Remix, sent to the newly created contract addrees 200 BAT tokens and tried to interact with BATSend by calling transfer on ERC20 interface and entered 20 in the tokenAmount parameter.

Expected output, my account will have +=tokenAmount and it didn't happened, tx was confirmed and account amount didn't changed.

First, as pointed by @goodvibration, you don't need to recast BAT_TOKEN_ADDRESS to the ERC20 interface since you already did it when assigning its value.

Second, your BATSend function should check whether the tokens transfer was successful or not since transfer returns a boolean (true if the transfer was successful, false otherwise), like that :

 function BATSend(uint tokenAmount) public payable
 require(ERC20(BAT_TOKEN_ADDRESS).transfer(myAddress,tokenAmount));
 

Now, concerning your balance not being increased, it's probably because your contract's balance is empty, or inferior to the amount of tokens you are trying to transfer.

Whenever you transfer tokens, you should be aware of how many decimals the ERC20 uses. In your case, BAT uses 18 decimals, so you should multiply the amounts of tokens you want to send by 1e18.