Upvote:2
Yes, you can.
Now if I had to do that I won't have enough time to get my UK visa before we leave.
They will not help an applicant with a tight schedule just because an applicant failed to apply in time unless there is an emergency.
You will definitely need both visas before you travel there (legally), so your best bet is to use Priority processing options and pay extra fee to get the decisions quicker. Although UK visas do not require 8 weeks, specially not from the US. You can check for your location here.
You also have an option of keeping your passport while applying for the British visa (you have to pay extra) . That way you can apply for both visas at the same time and send your passport for Irish visa and get the UK visa stamped after/if they call you for it.