Is it a council run allotment or a private one?
Our council are not great at answering phones either at the moment as a lot of them are working from home. They will however have a customer care policy that says they should answer the phone. This is a link to a page from our council’s website, but yours should have something similar if you click the Contact Us link at the base of their home page.
https://www.reading.gov.uk/council/contact-us/Scroll down a bit and it has a make a complaint on-line link highlighted in blue. If you use this, they will respond with an automated response, but should also pass it on quickly to the department you need to speak to.
Second thing is that a lot of standard letters are getting sent because people are not in the office to press buttons. We are getting final demand notices every month for rent money on the business premises we rent from our council even though it goes out by standing order and has done for years. Any queries are dealt with by a ‘oh just ignore those’
Get an email off to them as soon as you can, with pictures of progress made if possible. If there is asbestos, they should remove that for you in any case and probably the tyres as well. You need to read the tenancy agreement for that, but ask anyway and explain why it is still there. It may take more than 7 days to get a reply, but it does not mean your in-laws have lost it for good. Register your appeal within that period and keep in touch with the council over it.
Good luck with it all