As she is coming to the UK to marry you, she doesn't qualify under those categories - she's not here on business, she's not a tourist, family visitor or student. When she enters the country, she'll be questioned, and if she says she's coming to the UK to be married, they will deny her entry, as her visa is not for the purpose of being married. If she's denied entry, they would send her and her son back, and it could be very difficult for her to return to the UK. She will definitely need to obtain a fiance visa in Taiwan from the UK embassy. If she did come on the 6-month free visa and you were to get married when she's here on this 6-month visa, there would be a lot of questioning, and she could get denied for a spouse visa if they find out you were together before she arrived. They want you to hold a visa for the exact purpose of your visit. It's best to follow the exact procedure, because there's a much higher chance she'll be able to stay here.
Also, I would assume she's getting a one-way ticket to the UK, if she's planning on marrying you and living here, and the border agency would find it suspicious that a foreigner had a one-way ticket and was here on a 6-month business/tourist/family/student visa. To be granted entry, she would need a return ticket. With the fiance visa, she's allowed to have a one-way ticket.
I'm American and went through the same process to move here and get married. I didn't need a visa to visit the UK, either, but had to get the fiance visa to enter the country, since I planned on being married.
