Do you mean the fare was £2
cash?
By quoting a £2 fare I’m not entirely sure that you are quoting like for like here.
I do not ever remember paying £2 on an Oyster, or a bus saver.
My memory tells me that £2 was the premium cash fare.
In early 2000s I remember a pack of bus saver tickets which equated to 65p per journey, and £1 if you paid cash on the bus.
Then there was some sort of 2 zone system - 70p cash outside the centre, £1 inside the centre.
Then Oyster was £1. Then 80p off peak, £1.20 peak for a while. Then became the same fare peak / off peak.
The £2 you are quoting was, from memory, the cash fare to encourage Oyster use. It was not the regular fare any more than the paper single tube ticket is the regular fare.
This was in the run up to the street side ticket machines in the centre of London (yellow route numbers at bus stops) when cash ceased to be accepted.
Addition
I’ve just found the bus fare history from Londonist which is usually reasonably accurate.
Bus single fare: Oyster PAYG (peak)
2004: 70p
2005: £1
2006: £1
2007: £1
2008: 90p
2009: £1
2010: £1.20
2011: £1.30
2012 (original): £1.40
2012 (revised): £1.35
2013: £1.40
2014: £1.45
2015: £1.50
2016: £1.50
Source
https://londonist.com/2011/11/london-transport-fares-2000-2012