GUWonder makes a very valid point. You cannot sleep overnight at LHR without leaving the airside area. If you leave the airside area, you would need to hold a transit visa.

As the Lusaka-Heathrow flights arrive at LHR at 1800, you must therefore take your connecting flight later the same evening in order to use the DATV exemption.

If your flight is the next day, you would have to get a transit visa even if the connection is within 24 hours. That is clear from the e-mail that you received, and you have incompletely summarised the conditions of the DATV exemption in the bit of your post that I've quoted.

One way around this problem might be to fly via JNB because BA's JNB flights arrive in the early morning, thus allowing you to make a same-day direct airside connection.

Also, you must be able to through-check your baggage to the onward flight and stay airside for the whole connection process. If the airline will not through-check your bag onto the next flight, then you'd have to clear immigration, reclaim your baggage, clear customs to landside, and then check-in again. This would require a transit visa as it is not a direct airside transit.

If you're flying on BA to London and your onward flight is written on a separate ticket, there is only a limited number of airlines onto which BA will through-check your bags. (This policy is of general application and is not aimed at people who need visas.) So you need to arrange your travel accordingly.