A bit of an historical situation which can be confusing.
Some properties are registered with the Land Registry. Other properties are unregistered but are effectively memorialised in the Registry of Deeds. In connection with the latter you would collect a lovely big bundle of documents as transactions occurred. The vital thing was to make sure that each important transaction was actually memorialised in the Registry of Deeds. Look at the bundle of deeds you have for the house and you will probably see annotations on them confirming date (and probably time) of memorialisation of the particular act in the Registry of Deeds.
AFAIK most new conveyances are now compulsorily registrable with the Land Registry. I understand that is a formal process which includes proper mapping. If validly registered you then get a simple document identifying a folio number with Land Registry who then hold the details for that property.
Like you, I inherited a house as well as a big deeds bundle full of documents memorialised in the Registry of Deeds. I simply instructed a solicitor to register the Deed of Assent in the Registry of Deeds. We did that because we have utterly no intention of moving / selling so there was no point in spending time and money going through registration with the Land Registry. However, if we ever want to sell we will then probably need to jump through some formal hoops to enable us to convey the property.
Be sure to have the Deed of Assent memorialised with the Registry of Deeds and to get it back from your solicitor. Instruct your solicitor to do it as it the house is a major asset. Alternatively, if you think that you might be likely to sell in the near future you should discuss that specifically with your solicitor who might advise you to register with the Land Registry.