I'd encourage you to go ahead and go this year. You can always start university socially transitioned, and then correct the paper trail after the fact. Just tell professors in advance that your preferred name is what you go by. You may be able to work with them to be documented as male on paper, but I'm not sure what rules they have in place.
As an adult, you can change your name whenever you want. I'm not familiar with the process in Ireland. Do you use the deed poll like the UK? You don't need a reason to change your name, just the desire to do so and whatever it costs. I'm not sure how long the process takes there. Here, it took me about 2 weeks to get mine back in the mail.
Your best bet to find out about your wait time would be to call the therapist's office directly and ask what the current wait is. You could also ask on average how many visits it takes to get a referral for testosterone. I imagine it varies there just like it does here. Some people leave from their first or second visit with a referral, some people go for months before they're ready.
