Could you please send your CV and CL, maybe I can provide you a more tailored advice.
Agree with the comments below and would like to provide some details on how to get a reference / pass the screening after consulting events:
1) Leverage your own network - find friends or friends of friends who can practice cases with you or make you a reference.
2) Attend company events. Consulting companies do a lot of events both for graduates and experienced hires. Find the social network groups related to consulting or websites, subscribe to newsletters and stay tuned. Also, check if they have events in your University since you are a PHD.
While some of the events will be open to everyone, others will require a resume and a cover letter, so make sure to prep.
3) Talk to people at the events and send Thank You notes. After each event, there is a Q&A session where you can talk to consultants 1 on 1 or in a group. Ask for contact info or send a thank you note after the event ("I just wanted to thank you for visiting our University... It was especially interesting to hear about... Would be happy to keep in touch and apply in the nearest future.). Alternatively, you may use linkedin for that. If you are an experienced hire I strongly suggest to ask for a 1 on 1 meeting in a thank you note.
4) 1 on 1 meetings. All people like giving an advice. So don't hesitate to ask consultants for a career advice. Tell your story and ask how consulting fits into it
5) Mock interview Depending on where you are in your prep process you may ask a consultant for a mock interview. Consultants are very much opened to help even if it is a cold call e-mail. The main problem is a lack of time on their side. So don't be afraid to remind of yourself if the consultant has already committed but finds it hard to find the time
6) Talking to consultants who specialize in nuclear engineering. Chances to get referral will be higher. Try to find them via Linkedin / events