I've started to write this post about a dozen times, as I keep thinking of all kinds of legal caveats. I'll try and keep it simple. This is English and Welsh law only -- Scotland and NI have their own legal jurisdictions.
Present day illegitimate children can inherit real (ie land, houses) and personal (ie goods, money) property under either parent's intestacy**, but they need to prove paternity when inheriting from the father. (**Obviously if there's a will, there's no problem anyway there.)
In the past, only legitimate (or later, legitimated) children could inherit on a father's intestacy, and there were various laws in the 20th century about this so if you're writing about a specific period I'd need to know when so I could check. (But go back far enough and Welsh law was different -- if the father acknowledged the child, he/she could also inherit.) All children could inherit on a mother's intestacy, though, in relation to personal property definitely and I'm pretty sure on real property.
Titles are different. Few women had titles in their own right, for one thing. Lady Diana Spencer was only "Lady" because her father was an earl -- so if she'd married a commoner her children would have had no titles (in the same way the Princess Royal's children have no titles). If she did have a title in her own right, eg as Anne Boleyn was Marquess of Pembroke, then its devolution would depend on the grant, which might well state that only legitimate offspring could inherit. (Possibly not relevant to you, but some titles could also pass to females, eg the present Countess Mountbatten of Burma, as that title was created with special remainder to the Earl's daughters, but only their heirs male.)
If the estate was entailed, as often happened in the past, then it could only pass in accordance with the tail -- ie under the terms of the original deed, and usually those revolved around legitimate male heirs.
EDIT: sorry, my brain wasn't wholly in gear last night -- I've corrected a very stupid mistake re Mountbatten. And just to make it clear -- not all titles are hereditary, so eg ex-politicians who are Baronesses can't pass those titles on as they are for life only.