Hello,
it looks like that the visa-situation for china did not chnage the last 20 years. When I came to Kathmandu in spring 1986 I got the information from the chinese embassy that they do not issue visa for individual travellers, for groups only and I should join a guided tour to China. But nevertheless I managed to cross the border with a friend, because we had the possibility to get our visa from Hongkong! ( This is a special story abou the how ) What I want to say is, that you should have the visa for China before you come to Nepal! So you can apply anywhereelse and again do not mentioned that you are travelling on a bike and that you will go to Tibet! And normally with this kind of visa it should be no problem going to China via Nepal! You will not have these problems when you enter China from Pakistan via the Karakoroum-Highway. Because of the weather-conditions the border is open starting from May.
Please keep in mind when you cycle in Tibet that most of the time you are in restriceted areas, which means, that there is a slight possibility to get stopped by the police. This happens in general not at daytime but at night when you stay overnight in a small local guesthouse. Whether you will be stopped by the police or not depends on the local authority (police). It was my impression at the time I was in China, that is was really unlikely to get caught, because if the police stops a foreigner they would have work with them - and they do not like to have work. Only if the policeman is ambitious and wants to make career in his job, only then it may be happen to get stopped. Then you have to look a litle bit stupid and tell him ( in case he speaks a little english), that this areas you cycle through is not restricted for foreigners but the administration in Beijing decided some weeks (months) ago that Tibet is now free accessible for tourists. He cannot check it! And if it happens really you have to take the bus to the next open city and maybe to pay a fine.
Velomade