does any of the time I've spent on a WP count towards the 2 years to get ILTR?
No it does not. The Civil Partner visa will be for 2 years duration and can be applied for as soon as the Civil Partnership has been registered. There is no need to wait until the end of your current visa.
During the last 28 days of validity of that 2-year CP visa you will be able to apply for ILR.
Why did I ask about your UK immigration history? To see if it is worth you staying on the WP route even after the CP has been registered. On your facts it is not.
Do appreciate that once you are in a CP with a British Citizen that there is a 3-year qualifying period for applying for Naturalisation as British, rather than the normal 5 year period. So on your facts, as soon as you have your ILR in "2 and a bit" years, you will be able to apply for Naturalisation as British, because clearly you will have been in the UK for more than 3 years at that point in time.
Do apply for the CoA as soon as possible. Such applications take about 2 months to process. You and your partner cannot go along to the Register Office and give the need Notice until you have your CoA.