You actually cannot travel to the UK after spending 90 days in Ireland. The 90 days is for the whole common travel area, not just the Republic of Ireland!
The UK and Republic of Ireland form part of the common travel area
This means that there are no border checks, immigration, or customs between the two countries so there is no possibility to get your passport stamped as proof. Therefore, the 90 days stay which you were allowed when you entered the Republic of Ireland applies to the whole common travel area, not just the Republic.
This post describes in more details for different circumstances.
Implications in crossing the Ireland/Northern Ireland land border for an American
FYI regarding cross border travel: It is generally accepted that you need to retain your own proof of travel between the countries. This means boarding passes, ferry tickets, train tickets, bus tickets etc. that prove your travel. This should be sufficient in most cases.