Hi there good luck to you. Keep it up. Remember one slip doesn't ruin the whole project. A bad day, or a bad week, is simply that. It isn't an excuse to walk away and give up. I remember someone saying this isn't defined by speed or distance, its defined by direction - keep heading in the right direction.
My story is that at slightly under 5 feet 10 I slowly put on weight from my late 20s to late 40s. I only made it to 18.5 stone, but realised it was having a huge impact on my life, breaking chairs, being out of breath, being worried about exceeding weight limits for equipment. I began to lose weight, got down to just under 16 stone and found I was stuck, so I bought a bike. That changed my life. The bike became much more important than my weight. I lost about 5.5 stone, but it took me 15 months, at about a pound a week. Remember the speed isn't important.
More importantly perhaps, I'm now 10 years on and have kept most of the weight off. I'm still cycling, do long-distance endurance stuff and am fitter than I was in my 30s, at almost 60. So it can be done. I tried to run a calorie deficit Monday to Friday, and a balanced diet Saturday and Sunday, hoping I'd prevent a slowdown in my metabolism - almost a reverse 5:2 diet.
People who say exercise doesn't matter compared with calories in, are sort of right, if you do 40 minutes in a gym and think that is enough to justify a meal, but it doesn't apply in the same way to cycling serious distances. I can burn an extra 600 calories an hour quite readily on the bike. 10 hours a week, about 150-160 miles is 6000 calories. A pound of fat is worth about 3700 calories. If you are doing 10 hours a week of fairly strenuous exercise, without increasing your dietary intake that is worth almost 2 pounds a week. I'll put on half a stone in the autumn as my cycling slows down, and will take it off again in the spring, without really trying, as my mileage increases.
Anyway good luck. It can be done, but it needs a determined long-term commitment, however it is worth it.
Don't worry about the bike. Robust wheels with plenty of spokes will be fine. The most important thing is to take it out and push your limits, health and doctor permitting.