With British Airways, can I access the business lounge in LHR if my connecting flight leaves on the next day?

To add to JoErNanO's answer which is spot on in general, in this case you will probably have Business Class lounge access on the basis of your departing domestic flight, so long as it's all been booked on a single ticket. (There are no departures from Heathrow before 6am, so you'll not be able to get access to the Concorde Room on the grounds that you've arrived in First.)

Although BA's domestic flights only have one class, certain tickets still allow access to business class lounges - Business UK fares. In general, an international First or Club World itinerary will couple with one of these fares.

On the other hand, if you've booked as separate tickets then unless you explicitly chose a flexible Business UK fare, you'll not have lounge access.

How do you find out after booking? Take a look at BA's Manage My Booking:

Firstly - it should show a Lounges section if you're allowed to visit a lounge. It doesn't always, but it's usually pretty consistent out of Heathrow.

Secondly, you can click on More flight information:

At the very bottom, you'll see a Selling Class. You want this to be J or Y.

Your final option would be to drop an email or call to BA's You First - as you've got a bit of First class travel going on, you get a special contact to deal with it. I've (sadly) got no experience of dealing with them, but I believe they're good and knowledgeable.

The lounge access depends on what time your connecting flight leaves. My understanding is that you'll be allowed access to Business/First Class lounges if and only if the connecting flight leaves on the same day or before 6am the following day. Quoting from the One World webpage on lounges terms and conditions (emphasis mine):

oneworld lounge access for First Class or Business Class customers

Indulge yourself before departure in a wider range of airport lounges when you fly First Class or Business Class with any oneworld member airline.

• Customers flying First or Business Class have access to the equivalent class of lounge regardless of their frequent flyer status. First Class passengers may use a Business Class or frequent flyer lounge if a First Class lounge is not available.
• First Class passengers may invite one guest to join them in the lounge. The guest must also be travelling on a flight operated and marketed by a oneworld airline.
• Connecting between oneworld marketed and operated flights:
  • First and Business Class customers connecting on the same day of travel, or before 6am the following day, can access the lounge when travelling between an international long haul (a oneworld international long haul flight is defined as an international flight marketed and operated by any oneworld carrier with a scheduled flight time longer than 5 hours) and an international short haul or domestic flight (and vice-versa).
  • Lounge access will be determined on the international long haul ticketed flight (either First of Business Class) regardless of the ticketed class of travel on the international short haul or domestic flight.
  • You must be prepared to show your boarding pass or itinerary showing travel in First or Business class on the international long haul flight, in order to access the lounge before your international short haul or domestic flight.