In short : don't demand your girlfriend goes on birth control, but have a discussion about it.
I waited until I became sexually active to go on birth control. I wasn't comfortable playing with my hormones like that, but using condoms did get tedious after a while and I decided I'd rather use a contraceptive. With my doctor, we decided on one which met my needs. This was all my decision, from start to finish. I'm a pretty laid back person, but if my boyfriend at the time had said "isn't it time you start using the pill ?", I would have gotten angry. (Not saying that's how you were going to talk to your girlfriend, but I'm feeling a sprinkle of entitlement in your post that would not go over well, I imagine.)
You can absolutely share your worries, but I wouldn't demand anything of her. There are several reasons for that :
Her body, her choice
Using a contraceptive is a deeply personal decision. So first, don't expect your girlfriend to use a contraception. It is a possibility that you wish to discuss, but don't go in there thinking she should do it.
Second, don't suggest yourself what kind of contraceptive would be appropriate for your girlfriend. If she agrees to use a contraceptive, she gets to decide what would be better for her. Depending on her needs, any medical issues, side effects... this is a decision she gets to make, preferably with her doctor. You can voice a preference, what you would be more comfortable with, but that's it. Since women being on birth control has become a "normal" part of life, I think too many people think it's not a big deal (and thus, asking your girlfriend to do this is ok) and ignore how much pain/hassle/stress/... can go into that.
If you feel so inclined, here are some resources on what going on birth control can entail :
You can do (a bit) more
First, I want to address the reason why you want her to use a contraceptive : your fear she'll get pregnant, which may be irrational. You say there have been "isolated" signs of her being pregnant. Since you don't mention what those are, I can't really get more specific, but a lot of signs of pregnancy also occur for other reasons. So are you just afraid she'll get pregnant because she's not on a birth control (and it's a first for you, you're not used to it, your mind sees things or interprets stuff in a weird way), or are there other reasons ? Digging a bit there might help you find a better solution. Plus, if my partner wanted to talk about me possibly going on birth control (or modify my body in another way), the more he can explain the why, the more sympathetic I would be.
Also, you seem to think that female birth control is the only way to ensure no pregnancies. First, may I remind you that no contraceptive is 100% reliable (Planned Parenthood has a great website for this). And also, there's more you, the guy, can do. Sadly the man pill is still not a reality, so your options are quite limited, but they're there and I think being informed about them and truly considering them will allow you to have a more wholesome discussion with your girlfriend.
- Condoms with spermicide, which increases the condom's effectiveness
- The pull out method (only when combined with the condom though !)
- And if you plan to not have children at all, a vasectomy : very effective, very little risks of complications or side effects (way less than a contraceptive has for women)
Again, I understand that the options for you are limited, there's quite a bit of hassle there or you just don't want to "mess" with your own body. I just hope that you in turn will understand your girlfriend's reluctance to do the same and show some sympathy.
Have a discussion
With all this information in mind, I hope you can have an open-minded and constructive talk with your girlfriend. Since you've already talked about contraception at the start of your relationship (kudos for that !), and she wasn't completely closed to the idea, I think this will go well as long as you don't go in there demanding she goes on the pill or get an implant. Explain your concerns, and ask her if she has given any more thought to going on birth control and if she would consider it more seriously now.
As for the ban on sex until all is secure, depending on several factors (the time to get an appointment, some birth controls can only be started on a specific moment of your cycle, some again aren't effective immediately and have a waiting period), this could range from a week to a month. I understand if you're uncomfortable with penetrative sex until a solution has been found and put in place. But there's a lot you can do still ! No need to put a ban on sex all-together, just the part where pregnancy is possible. Hands, oral, buy toys, get a bit creative,... Or use the combo condoms/spermicide/pull-out method. I think this will go over much better with your girlfriend then just saying "I'm not touching you for the next two weeks".
Last side-note : don't know what your finances are (and hers) or who pays for condoms, but consider paying for half of her birth control.