There should be no need to follow up by post.
I have appealed a Smart Parking charge myself somewhat recently (last 6 months) and from memory, their acknowledgement email came a couple of days after submitting, and their response a few weeks after that. Keep a close eye on your spam folder, the first email I had from them went into my spam.
The Code of Practice requires them to respond to appeals within 28 days, so if you haven't heard anything from them by such time, then chase them.