I almost entered Israel from Egypt using the Taba/Eilat border crossing Michael Seifert mentions in his answer in 2016. In that year, there was a local bus connecting central Eilat to the crossing; I strongly assume that it still exists because it also connects places such as the underwater observatory to the town centre. So on the Israeli side, there should be no problem getting into town and getting onwards. Note that if you plan on renting cars make note that you will very probably have to rent a different one on each side of the border.
Israel and Egypt have signed a peace treaty and otherwise have diplomatic relations with each other; much like Israel and Jordan. So there are no difficulties when travelling between those three countries. Note, however, that having an Israeli stamp in your passport will stop you travelling to a number of countries. In the eyes of these countries, even an Egyptian exit stamp from Taba will be evidence that you have visited Israel making you unadmissable in their eyes. This is less of a problem to you because you have two nationalities and hence two passports. I would strongly recommend you to choose one only for the trip and to make sure you don’t intend to travel to any of those countries on that same passport. Other than that, I am not aware of any different treatments of British and Irish citizens in Egypt or Israel that would tip the decision in favour of one or the other.
Finally, I wish to come back to my first sentence in which I said almost. In the end, my group’s trip which was planned to be Jordan–Egypt–Israel was rerouted to become just Jordan–Israel. The reason for this being the presence of terrorist groups such as Al-Qaeda and IS on the Sinai peninsula. I am not sure what the current situation is like, but I strongly suggest acquiring as much information as possible on the Sinai peninsula from travel agencies and/or government sources. For example, the German foreign ministry has issued a travel warning for the Sinai peninsula (in German). This travel warning explicitly states that one should not attempt tours on one’s own. (‘Von Überlandfahrten ohne ortskundige Begleitung wird dringend abgeraten.’) Likewise, they explicitly recommend not to visit any parts of the peninsula other than the East coast strip (‘Von Fahrten abseits des Küstenstreifens […] wird gänzlich abgeraten.’).
I myself am German, so I knew immediately where to find German sources and recommendations. Since you are a British citizen, the advice given by the UK government and found in Tim’s answer along with the links therein may be even more relevant. While details of the recommendations may differ, the general picture of the advice should stay the same.
Of course, it’s your choice in the end but do consider that it may be more than just rough.
