You could do both. If the tiara can be worn underneath the veil, you could wear the veil on top - this would cover your face when you walk down the aisle, then when the veil is pushed back off your face after the ceremony the tiara will be more clearly visible. If you are having a reception afterward, even a small one, then you can opt to remove the veil entirely in between the wedding and reception. This is what lots of brides are doing these days (at least in the US) - that way you can best show off your hairdo, or tiara, or whatever is under your veil.
About your concerns: if you are wearing a traditional style of wedding dress (which it sounds like you are) then a veil and a tiara would not be 'too much' for a registry office wedding. It is your wedding day, it is special, and you should do it how you want to. Besides, it may 'complete' the overall look (not just for you, but from the point of the groom and witnesses too). However, that is not to say that you absolutely need to wear a veil - it is really a personal decision. Go with what you think looks best for you (& what the groom will like) - you have so many options whether you do veil only, tiara only, tiara + veil, keeping veil on all the time vs off at the reception - and they are all GOOD options so don't worry. Just pick what you like best and enjoy this exciting time of your life!