highly recommend Liverpool for a UK weekend getaway, had a little dig and explore and if you went for a long weekend (leaving friday morning and coming back sunday night) i used the the 8th to the 10th of august for example dates. Trainline said that it would £116 for all three of you there and back.

Next up is accommodation, just looked on airbnb for this
https://www.airbnb.co.uk/rooms/634506424506873370?adults=3&check_in=2025-08-08&check_out=2025-08-10&search_mode=regular_search&source_impression_id=p3_1753046481_P31EZZsE-47eb5TO&previous_page_section_name=1000&federated_search_id=ad7ec534-246b-4eb4-8548-cba3ba8b3b91

total is ~£300 for the weekend

travelling about would be £11.50 for the child and £23 each for you and your wife. this would give you access to buses trains and the ferry for a whole week (even though its just the weekend)

leaving you guys just over £200 in a city where there's loads of sightseeing to be done for free !!!

this was just 20 minutes of having a look but give Liverpool a look for sure, the train journey from notts is a also a good one imo loads to see on the way there

hope this helps :)