Ooo nice buggy that's the one I always envy when I see it. Is it fairly light and easy to manoeuvre? If so I think it would be fine on bus tube etc.
Can someone help you onto train at your home station? If not I usually find that people will help lift buggies if needed, but most of them can just be tipped up on back wheels and onto trains.
I don't often use Covent Garden tube so while I remember there is a lift, I can't exactly remember if there are small flights of stairs to the lift. If you're happy bumping buggy up and down a few steps it would be possible.
However, Neither Paddington nor Waterloo have fully step free access to tubes I think. I find it easy to go up and down on escalators (and can try to explain how I do it if you want )but if you're not confident with them then I'd suggest a bus.
Buses are pretty easy in Central London if you're not in a massive rush (so you can wait if a bus goes by and the wheelchair space is full). Have seen plenty of women with your buggy on buses, so is doable.
from Paddington the 159 will take you almost all the way to Covent Garden but will take an age.
From Waterloo lots of buses will go towards Covent Garden, again you will have to walk a wee bit.
TFL website lets you specifying hat you want buses only or step free access in the journey planner, if you do go for buggy.
If you go for carrier can you keep weight down by only packing some milk and picking up cartons of pre made stuff when out? There's a boots in Covent garden which would probably have some.