re :I've been downloading them to my PC but in the meantime they all have been removed from my email client/account.

Thunderbird is the email client program, so perhaps you mean your webmail account which you logon via a browser.

It sounds like you are using a POP mail account as removal from server upon download is the common practise unless you set instructions to 'Leave messages on server'. So I would set this up now. Right click on pop mail account name in Folder Pane and select 'Settings' Select 'Server Settings' for example see image below.

The only way you can put emails back onto a server is by creating a new existing mail account for same email address and set it up as imap. Then you have folders that can synchronise with server. Then use right click on highlighted emails and select 'Copy to' and choose imap account folder. Do it in batches and check webmail to see they are on server. I suggest 'Copy to', so that you do not lose anything from the pop mail account.

But remember imap mail accounts show what is held on server, they are effectively one and the same. Anything you do in imap account will effect what is on server and vis versa. So if you access webmail via a phone and delete an email off server, when Thunderbird connects that email will get deleted from the folders in the imap account. Once complete, you can either choose to keep the imap account or remove it.

Sorted! Thank you for your help, much appreciated!