After arriving at the mainline terminal station from your origin, you will be able to use your ticket for a single journey to a London Underground station anywhere in zones 1 or 2. This includes Old Street (which is in zone 1).
After exiting a London Underground station, your ticket will no longer be valid (and the barriers will mark it electronically to indicate this).
Tickets to London Underground destinations from outside of London are grouped together into "zonal" destinations, as you've found. This reduces the number of different fares that the ticketing system needs to accommodate, and also offers some added flexibility to the traveler. (Travelling back out of London, it may be more convenient to start your journey at another station in the same zone such as Barbican or Angel rather than walk to Old Street, or pay an extra fare.)
Ticket prices for travel to a zonal destination in London are calculated by starting with the fare from your origin station to "London Terminals", and adding on a fixed rate according to the zone (£3 each way for Zone 1-2 as of 2019).
In the case of Old Street and Moorgate, because they are considered to be London Terminals for travel from the North, and they have had a mixture of Underground and National Rail / British Rail services over the years, there is a historical entitlement to travel by National Rail to King's Cross, then continue by London Underground to one of those stations. This rule is not well documented in publicly available material, but here is the description given in Railforums UK's Ticketing Guide:
7.3.3 National Rail tickets valid on Underground/DLR
7.3.4 Validity to Moorgate and Old Street
If a National Rail ticket, with a destination/origin of London Terminals is valid into/out of King's Cross, then it is also valid for travel for a direct journey between King's Cross St. Pancras and either Old Street or Moorgate only. Break of journey is not permitted.
On this basis, if King's Cross station is a valid terminal for your journey (which is likely as you're travelling from the "north of the UK"), a ticket to London Terminals would be sufficient, as you technically don't need the £3 supplement for travel to a Zone 1-2 station. The journey planners used by online ticket retailers tend not to include these inter-available routes in their calculations (see discussion on Railforums here), so they will add on the Zone 1-2 supplement anyway.