Morally I think you're in the wrong recording people without their consent. But then an individual's moral standing or failing is their own business, and everyone's is different.
Legally it's a grey area, and is subdivided into data protection considerations, and privacy considerations. You have made it greyer still by a) including sound in the recording and b) not informing people in advance that they are being recorded.
Data protection and privacy laws do apply to private individuals too. Although the Data Protection Act 2018 (which mirrors GDPR, even though GDPR no longer applies in the UK) largely covers recordings outside a domestic boundary. This might cover the doorbell cam (depending on the angle and how much outside your boundary it records), but not the puppy cam. So the ICO isn't your primary source of guidance, actually.
The law is murky on this, but the basic tenet of domestic CCTV centres around you operating the system in a responsible way that protects the privacy of others. If questioned, could you demonstrate that your recordings are a) responsible; b) reasonable; and c) protective of the privacy of others?
Basically, you are allowed to record your own conversations in your own home. If you're recording and you are not there in person i.e. on the footage and other parties haven't given consent beforehand, I think you're potentially on shakier ground.
Domestic CCTV is covered and regulated by the Protection of Freedoms Act 2012. This is more to do with privacy laws than data protection.
You are entitled to record to 'protect your property'. And I'd argue that it's not unreasonable for an estate agent to expect to appear on doorbell cams, external CCTV, nanny cams and puppy cams from time to time.
However, you have potentially violated the privacy rights of the viewers of your house by recording them without their knowledge. Especially if the footage includes sound recording. They have privacy rights, even within your house. And possibly data protection rights, from the doorbell cam (depending on how much footage outside your domestic boundary it covers).
And if the recording could be considered to be harassing, anti-social or intimidating, that could be a criminal matter.
Furthermore, because you have instructed an estate agent to conduct business on your property, and they are carrying out viewings as part of their job, the lines between private and public/commercial become further blurred. This might open you up to some of the data protection requirements that you believe to only apply to businesses. (This one's a grey area too, I believe.)
If you want to continue recording people, I think you need to be very clear upfront that this is what you're doing. Then estate agents and viewers alike can make an informed decision about entering your property. That would also be the moral thing to do.
I wouldn't care that you had cameras running in your home as long as I knew about it in advance. If I found out later that you'd been recording me without my knowledge or consent, I'd be extremely pissed off, and would probably seek to make a privacy violation complaint.
If you think the estate agent has acted improperly, just change estate agents. You have the freedom and the right to do this. Be clearer with the new one that your puppy cam and doorbell cam will be recording them, and by extension viewers. I think that if you produce the footage as proof of a poor job to the current estate agent then you will open yourself up to potential legal action. The law might ultimately come down on your side, but you'd likely have to spend a lot of money arguing that case. And who needs the hassle when you're trying to sell a house?