I've been through an annulment myself and help a lot of people through the process as part of my job in the church.
If you wanted to participate, it would make it a whole lot easier for your ex. I was lucky that my ex agreed to take part in it and so we were both interviewed (separately). We also provided one witness each. He chose his dad and I chose a friend. The witness, if you provide one, should be someone who has known you well both before and during the marriage.
If you chose to participate, you would go along to the diocesan offices (they can probably also do it remotely, depending on the diocese), not to your local priest. It will be a canon lawyer who asks the questions and makes a statement out of them, statistically likely to be a woman, if that makes any difference to you. Yes, the questions are very instrusive, but I actually found it quite healing to think through everything in depth and consider what had really happened in the marriage.
There's no blaming anyone in the process. The lawyer who takes your statement doesn't pass any judgement on what went wrong and why. They're just looking to see whether there's any evidence that would fulfil one or more of the grounds for an annulment, in the UK it is nearly always the argument of a 'lack of discretionary judgement'. You don't go up in front of a court or get cross-examined or anything like that, just sit in an office and talk about the relationship as a one-off.
As a worse case scenario they could give a judgement saying that the person can't remarry in a Catholic church even if the annulment is granted, either forever or subject to certain conditions (for example if they were abusive), but that's rare.
If you choose not to participate in the process, they can do it without you, but it's much harder to get the proof and it often fails or takes a long time to process (searching for extra witnesses etc.). You're under no obligation to take part. You also have the right to choose not to answer certain questions if you'd rather not. You're there voluntarily so you only have to do what you're comfortable with.
As pp said, it in now way affects the legitimacy of children. The Church no longer has a category of legitimate and illegitimate children in any case, it's not a matter of canon law. But because you (both) believed yourselves to be married at the time of the conception of any children, they would have been considered legitimate even when the Church did have those distinctions.