Free Traffic Legal Advice

I'll try to keep this as short as possible:

• Car was knackered, decided to have it scrapped.
• Forgot to renew my vehicle tax on 1st April 2023 (which would have been £35 for the year).
• Car was eventually scrapped on 12th June.
• New tenants at old address forwarded mail today - which includes penalty for untaxed vehicle dates 30th June, and a Single Justice Procedure Notice issued 17th October with 21 days to respond - which has now also passed.

I'm panicking a little here, I am totally at fault but don't know what happens next as I've missed the deadline to respond to the SJPN. Do I end up with a criminal record because I forgot to pay £35 tax? Can I appeal in any way?

Pay the (small?) penalty, including back-tax and move on IMO

But it's not quite that simple. He has a SJPN informing him of court action. The "small" penalty the DVLA might offer is now off the table.

You should respond to the SJPN asap pleading guilty. The deadline was only last Friday so you should be OK if you can respond online. Unless the DVLA can be persuaded to settle out of court in the meantime (and you'll have to contact them to find out) you will face an income related fine, a surcharge of 40% of that fine, a contribution towards the DVLA's costs and ordered to pay the back tax.
If the DVLA is uncooperative You could, when responding to the SJPN, request a hearing in the normal Magistrates' court. You could attend that hearing and ask the prosecutor whether  an OOC settlement can be made.