There is no list of questions or subjects that employers are not allowed to cover in an interview. Employment law just isn't that prescriptive.
However questions about children or personal circumstances may indicate discrimination if they (for example) are only asked of female candidates, or if there is reason to believe they have made decisions based on the answers given.
For that reason, although not illegal, it is daft of any employer to ask that kind of question, because even if they do so innocently, it leaves candidates with the impression that they might be discriminated against, either during recruitment or at work, and it means recruitment decisions are more likely to be challenged.
OP it may be that there was no malicious intent with those questions, particularly as you were wearing an engagement ring. They may have been trying to be friendly or put you at ease. Or they may have been trying to suss out how much support you might have if you have children and how much time you might therefore need off.
You'll have to decide which you think it was, and from the overall impression you got, whether you'd still like to work there.
They can ask you those things I believe as long as they ask all candidates the same questions...but you should probably ask yourself why they asked all that stuff which can't possibly have been relevant to your ability to do the job -- at the very least they must lack personnel training. I've been asked illegal/dodgy questions a few times and have generally seen it as a clue about what it would be like to work for the people asking them...
I think they can ask whatever they want including about plans for DCs, childcare etc - they're just not allowed to discriminate on the basis of your answer (so most companies don't even ask as then how do you prove that they didn't use that info in reaching decision?). But I am not a lawyer, just read a lot of employment threads!
I was asked when/if [can't remember which] I am getting married. I know they can't ask about children or plans to have children, but are they allowed to ask about plans for marriage? It may have been that they saw my engagement ring and were making conversation, but either way I'm curious.
They also asked other things that I thought weren't relevant to the situation, eg. what my DP does for a living, where he works, where my family live and quite a lot of other things.