Every time there's a strike in Israel, which is pretty often, Americans get to reminiscing about how Reagan dealt with with air traffic controller strike in 1981 - he ordered the striking controllers back to work, and fired those who didn't do so within 48 hours. It's a nice fantasy to imagine something like that working in Israel, but I think that the culture and bureaucracy are too different from those in America for such a move to work and not backfire politically.
The current strike in the Foreign Ministry is unique, in that it's not just Israelis who are being affected. Visits by foreign leaders are being disrupted, and Aliyah visas are not being issued. The latter is affecting those who are planning to immigrate this summer. Nefesh B'Nefesh, which aids and helps process olim from North American and England, estimates that hundreds of potential olim will be affected if the strike is not resolved.
However, there is a simple solution to the problem. A strike can only work if the striking workers have the power to stop some service. Reagan showed that the striking air traffic controllers did not have this power by firing them and finding replacements. Nefesh B'Nefesh, as a representative of potential olim, can make a similar end-run around the Foreign Ministry.
There are two ways to make aliyah. One is to get an Aliyah visa from the Israeli Foreign Ministry while still in your country of origin, and obtain citizenship as you enter Israel. The other way works for those already in the country. You can go to the Interior Ministry in Israel (which, as far as I know, is not on strike), and they will process your Aliyah.
Instead of dealing with the Foreign Ministry, Nefesh B'Nefesh should go straight to the Interior Ministry here, and work with them to process the hundreds of people they will be bringing. These olim-to-be can fly to Israel on the planned charter or regular flights, enter the country as tourists (like any North American Brit does when they visit), and be immediately processed by Interior Ministry officials. So instead of entering the country as olim, they will enter as tourists, and then become olim a short while later.
There are some issues with this plan which would need to be addressed. The flights to Israel are payed for by the Jewish Agency. They have always (by always, I even mean pre-NBN) paid for the flight to Israel for anyone with an Aliyah visa (at least from the US; I don't know it this is true for olim from other countries). The Jewish Agency would have to be convinced to pay for the flights for planeloads of people who will enter as tourists and make Aliyah soon after their arrival. However, the Jewish Agency's head of Aliyah, Yehuda Sharf, told the Jerusalem Post that the agency "intends to do everything in its power to alleviate their problems". They may have reservations about the plan, but since they were going to pay for the flights anyway, overcoming them shouldn't be much of a hurdle.
I don't know what other issues may arise, including legal ones, but there's at least one member of Knesset who made Aliyah with Nefesh B'Nefesh's help. I imagine he's be more than willing to assist in any way he can.