I can't give you the wife's perspective, but I can tell you about my husband's experience when he was in almost your exact same position about 18 years ago. He was the biological son who showed up unexpectedly. The main difference is that his bilogical father ("biodad") had recently gone through a nasty divorce and custody battle. When my husband was 30 he called his biodad, who lived across country, out of the blue and introduced himself. They made arrangements for my husband to fly out there, meet him and two of his other three children. The weekend had its ups and downs, with the biodad not really ready to accept full responsibility and start a relationship. One of his other sons was accepting and one was very distant. Biodad's ex-wife and other children had no idea of the possibility of my husband's existence. My husband was the result of a brief college affair years before biodad met his ex-wife. After that weekend they kept in touch a little, but eventually lost contact again. Fast forward 10 years right around my husband's 40th birthday and he tracks down biodad again, who has moved to the other side of the country (we are in the middle). This time they start a little more gradually with some phone calls, cards exchanged, etc. Plans are made again for a boys weekend in a neutral location with biodad, my husband and biodad's other two sons. At this point, his ex-wife and his daughter still don't know about my husband. This weekend goes much better than the weekend 10 years previously and they all begin to keep in touch. My husband discovers he has a lot in common with one of his half-brothers, the one who had been initially more accepting of him. The oldest brother is still slow to warm, but eventually comes around. In the last 8 years, the relationships have grown, the ex-wife and the daughter were told and biodad is attempting to establish a relationship with our sons, his only grandchildren at this point, although it's challenging because of the history and the geographic distance. I have no idea how the ex-wife reacted to the news, but she was gracious (at least to our faces) when we eventually met her for the first time. The daughter, who is close to her mother, has been very accepting and has dinner with my husband whenever he travels to her part of the country.
We all took a big step forward about a year and a half ago when the half-brother with whom my husband had an instant connection invited our entire family to his wedding. Biodad comes from a big family and we met all of his remaining siblings, their families and the ex-wife and many of her friends and family. Some of have been more accepting than others and I suspect our presence at the wedding caused a little stir, but everyone was respectful to us in our presence.
As a sad footnote to this story, biodad was diagnosed with Stage IV lung cancer in August. My husband and the initially slow to warm half-brother were both at the hospital with him during testing and diagnosis. He is undergoing treatment, but has only a few years left at best. J. as the relationships were deepening we are going to lose him. My husband spent several days with him in September helping him get his house ready for the market and helping get some other affairs "in order." We were all invited to join them the week after Christmas to continue their holiday celebration, but our high school son has some committments that week we can't avoid.
My advice would be to take it slowly and let your biological father dictate the amount and type of contact. This is a big shock and adjustment to both he and his wife and any children they may have. In retrospect, we are glad my husband tried the second time, but I don't think he would have ultimately have been so successful without the gap of time in between for biodad and his children to process and become more accepting of the situation. My husband has also been very understanding about the fact that his biodad essentially abandoned him when his mom was pregnant. They did not have a serious or long term relationship in college, but if I were my husband I'm not sure I could have been quite so forgiving and accepting. Sorry for the length of this response, but I thought you might appreciate hearing from someone who's been there. Best wishes.