Why do you say it is rarely important? curious to know
Registering the ltd company name doesn't mean much - You could register Singlemum Ltd, and I could register Singlemum UK Ltd, SInglemum Worldwide Ltd, Single Mom Ltd, Singlemum Sales Ltd, Singlemum Consulting Ltd, etc etc
Further, I could register Maxwell Holdings Ltd and simply trade as 'Singlemum' anyway. Customers rarely care for the ltd company name and usually only see the trading name.
The best protection from someone like me trying to trade as Singlemum is to register a trademark.
Without a trademark, you might be able to rely on passing off (me trying to pass myself off as you), but that would rely on you having some established goodwill in the name 'Singlemum'. You won't have any goodwill in the ltd company name, because it is dormant and doesn't trade. So your goodwill will only ever come from your sole trader name anyway. So having the ltd company name doesn't really help you.
With a trademark, its much easier for you to stop me trading as Singlemum.
If you ever want to go limited, you don't
need your company name to be Singlemum Ltd, for the same reasons outlined above - you can be Jane Doe Ltd trading as Singlemum. It might even be helpful to keep Singlemum as just a trading style, under a more generic company name, in case you ever decide to expand and add more brands/trading styles later.