How can they link your data if you delete your account and sign up again with a different email?
With ease, if they actually wanted to. It would be treading a very dangerous data protection and computer misuse line, but a website which wanted to track users across registrations would have many ways to do it. All can be defeated, but would work for I suspect 90% of people likely to be signing up for a forum.
Firstly, they can use cookies. When you log on to a website like mumsnet, authorisation is done using a cookie. A cookie is a small piece of data which is supplied to the website which gave it you every time you visit a page on that website.
So to login you supply your username and password to one page, and the site gives you back a cookie. It probably looks like dDaQU/lerSASBo/+TwYfuA==. Sometimes it encodes some information (ie, you can look "inside" it and it means something), more often it's just an encoded random number (as that is).
A database is then used to say "anyone who presents the cookie dDaQU/lerSASBo/+TwYfuA== when accessing this site, for the next 72 hours (or whatever) can be assumed to have logged in as user so-and-so".
So if you delete your account and then create a new one, the cookie from your last login session may well still be passed across. If the site is more nefarious, it could actually drop a long-lived cookie for the express purpose of doing this, but it's more likely to happen "by accident".
OK, you say: I'm across that. I'll delete my account, delete all the cookies from my browser (it isn't enough, for various reasons too tedious to go into, to just delete the cookies dropped by MN), and then create a new account. Result!
Now we're off into the world of "MN don't do this, I assume, but they could tinfoil hattery". Browser Fingerprinting is the technique of looking at the precise configuration of your browser: not only the version, and the version of the operating system you are using, but the fonts you have installed, the extensions you have installed, any number of other subtle "my browser is not quite like your browser" issues. Combined with, say, the IP number - it won't usually uniquely identify a user, but it's likely to remain constant over a "delete my account, create a new one" session, and even if it doesn't, the particular ISP you are using is likely to remain the same - and you have a good chance of identifying delete and re-sign pairs. Not always, but often. Or at least sometimes.
I would, for the record, be amazed were MN doing this deliberately. But I wouldn't be completely surprised if they were collecting in their logs and trace information sufficient that a bad actor with access to data collected for good purposes would be able to re-analyse that data to spot such pairs. A capable threat actor with administrative access to a system is not to be trifled with. Which is why controlling, and indeed filtering, logging is a serious issue in overall security.