Your scenario is not completely clear, because you don't tell us whether the attacker is producing valid blocks or invalid blocks. You have also not defined whether your own wallet is a full-node or a thin client.
Let's first talk about how blocks are relayed through the network:
When a new block is discovered, a node will announce via an
inv (inventory) message that it has this new block. Other full nodes will send a
getdata message, requesting the whole block. After receiving and validating the data, they will send an
inv message to their peers to announce the new block to them.
SPV nodes do not request the full block. They'll just get the block header, and if there are any transactions of interest, they'll specifically request proof for those to be sent via a
merkleblock message. SPV nodes cannot fully verify the validity of new blocks, because they don't store the blockchain.
Now there is three different possible scenarios, and two types of wallets to consider:
Attacker is producing invalid blocks
A SPV wallet may be fooled, because it can't verify the validity of a block. They still check that the block header's are well-formed. Some SPV wallets require multiple peers to tell them about the same block. Therefore, this attack may require additionally a Sybil attack or control of the SPV wallet's internet connection.
Full nodes are not affected, because they'll reject invalid blocks.
Attacker is producing valid blocks at low speed
A SPV wallet will reorganize to a longer chain when it hears about it from a different peer. So, again, the SPV wallet needs to be isolated from other information for this to work. As the information is valid, this attack may also work against full nodes. The attacked may notice the suspiciously slow update rate.
Attacker is producing valid blocks with 51% of mining power
The attacker is faster than the whole rest of the network. By choosing to only build upon his own blocks, he can mine 100% of the blocks and still outpace the remaining network. The blocks are valid and therefore accepted by all nodes (until some sort of intervention). The attacker can censor transactions and even rollback a small number of blocks by starting from an earlier block but eventually overtaking the natural block chain tip. This is called a majority attack and breaks the axiomatic security assumption of Bitcoin that more than half the mining power is honest.
How secure are six confirmations in an attacker's fork?
If you're caught in a Sybil attack, this will fool you. In every other case, you'll not be fooled, or the whole network is the target.