Welcome and congrats on making the decision to commute by bike!
I do 13mi each way commute daily, which is a comfortable distance, however my legs did feel it the first month or so when I increased the distance from the 5mi each way I did before. As a new commuter it will take some time for you to get your fitness (forget weight, it's not really relevant for commuting - it will come off as a byproduct), and I think you will struggle with a 44 mile round trip. That's a lot of miles for someone unused to regular cycling - I'm not trying to discourage you at all - but you do need to be realistic.
At that sort of distance, you probably need to think about working your way up to it - start off with 2 days a week and then go up from there. You also need to consider the facilities you have at work. Do you have secure cycle storage? Lockers? Showers? Can you access the cycle to work scheme?
It is also worth asking in a bit more detail about routing as the best route for cycling may not necessarily be the most direct - plenty of people commute into London on this forum, so if you give us a couple of postcodes near to the start/finish of where you are going (not the actual postcodes though), they will be better able to advise.
In terms of equipment for all year round commuting, you absolutely need mudguards and I'd suggest a rack and pannier bags as well although a rucksack can work well if you don't mind a sweaty back - I use one, but I have showers and changing facilities at work. At least one heavy duty lock is needed - if you can leave one at work then great as you can get something like an Abus granite X and leave it there, meaning you don't have to carry it with you all the time.
Also you need to consider cycle clothing - even with mudguards you are going to get wet feet when it rains heavily so spare shoes is a good idea as well as waterproof jackets and shorts/trousers.
In terms of the bike, the type of bike that you get almost doesn't matter, anything you get that is serviceable will do the job perfectly well, it just becomes a matter of what you find more comfortable. I much prefer a road bike to a hybrid for commuting, simply because of the greater number of hand positions meaning I don't get stiff or sore - but you may find that you don't like it at all. I wouldn't consider anything too expensive for commuting either, so an aluminium or steel framed bike is perfect.
I'd also look at e-bikes for commuting that sort of distance - especially as you will be new to it, a good e-bike will have plenty of range for you to do that sort of distance daily, and will make it noticeably easier for you to do. The other option would be a brompton - as this would allow you to take the train easily if you are feeling under the weather or have a puncture and are running extra late.