Can we take a rental car from England to Ireland?

You would need to check with your prospective rental car company, but generally, they do not allow you to take a car on a ferry. As an added hindrance, renting a car in Ireland has some added twists with insurance, making a British company more hesitant.

Many car rental companies have restrictions on some Eastern European countries AND Northern Ireland.
You might check if you pick the car up on London if there are any drop off fees in Scotland. If not, then fly from E'burgh over to Ireland on one of the budget air carriers and rent another car there. It's probably a more efficient and cheaper way to travel.

Did a quick check on the Hertz website and apparently there are no restrictions on taking a vehicle originally rented at Heathrow across on the ferry to either NI or the Republic. Would assume that other rental companies follow suit.
There is a surcharge for the privilege, however, as well as an international one-way drop charge if you drop it in the Republic, ie Dublin. Not sure about NI, where the vehicle would at least be staying within the UK.
You'd really need to check with the individual company to verify their policies, but the point is that there appear to be no restrictions.
Given the degree of hassle involved you're probably best advised to just fly over to Belfast or Dublin and rent another vehicle upon arrival.

Also there is often an extra charge for taking a car hired in the Republic of Ireland into Northern Ireland - rather less common the other way round. Always check the agreement especially if it seems a bit cheaper.

Thomas - I would contact prospective rental agencies and ask them.

Depending on your itinerary and how many are in your group you may find it more convenient -- and cheaper -- to fly and rent another car in Ireland. There are many flights to choose from and it can be quite inexpensive.

That Hertz reference is hopeful. They list a bunch of countries where you can't take their cars and Ireland and Northern Ireland aren't listed. I will call them, though.

We rented with Avis and Europecar and did that trip. Not a problem. We dropped it off in Belfast and flew back to London. There was a small one way fee and it was the UK and not Ireland. But is was worth it.

You should compare prices of two rentals vs the one way and the drop off on the other side of the Irish Sea. It really isn't difficult to do two car rentals on a trip. So, don't let that be the deciding factor.