Having starting the day on 88 for one, in pursuit of Sussex's 171 all out, the hosts closed on 443 for four, with Lumb scoring 171 and Taylor unbeaten on 106.
Riki Wessels (43), Samit Patel (43) and Adam Voges (32 not out) were also among the runs as Nottinghamshire finished the day with a massive 272-run lead over Sussex.
Warwickshire, who started this round of fixtures with a one-point lead over Nottinghamshire at the top of the Division One standings, also shone with the bat in their match with Surrey at Edgbaston.
The hosts progressed from their overnight 322 for six to 571 all out, with captain Jim Troughton making 119, Chris Woakes an unbeaten 118, and number 10 batsman Jeetan Patel 76.
England prospect Woakes then took the wicket of Zafar Ansari (10) before Surrey recovered to reach 109 for one at stumps, with Rory Burns on 53 not out and Arun Harinath unbeaten on 44.
At Chester-le-Street, Middlesex slumped to 58 for five by the close in pursuit of a victory target of just 118 against Division One strugglers Durham.
In a low-scoring match, Middlesex moved from an overnight 144 for seven to 185 all today - earning a first-innings lead of 83 - before then dismissing Durham for 200 in their second innings with Steven Finn (four for 80), Tim Murtagh (three for 41) and Toby Roland-Jones (three for 54) sharing the wickets between them.
That left Middlesex as firm favourites to take the victory, but they lost openers Sam Robson and Chris Rogers without scoring and also saw Joe Denly (16), Dawid Malan (15) and Eoin Morgan (two) dismissed as they slumped to 38 for five, Chris Rushworth taking three for 22.
John Simpson (10 not out) and Roland-Jones (15no) stopped the rot with an unbroken 20-run stand for the sixth wicket, but Durham will still feel they are in with a chance of claiming their first win of the season.
Kent captain Rob Key and opening partner Sam Northeast scored centuries as Hampshire's bowlers were on the wrong end of a drubbing before fighting back with a flurry of late wickets in their LV= County Championship Division Two match at Southampton
Key (119) and Northeast (140) compiled an opening stand of 252 on an easy-paced Ageas Bowl wicket before Hampshire claimed five late scalps to leave their opponents on 291 for five at stumps, James Tomlinson claiming three for 45.
Hampshire had earlier been bowled out for 292 without adding to their overnight total.
Yorkshire, who are fighting for promotion along with Hampshire and Kent, are in a good position against Leicestershire at Grace Road thanks to centuries from Adam Lyth and Jonny Bairstow.
Opener Lyth made an unbeaten 159, while England international Bairstow contributed 118 as Yorkshire reached 336 for five.
That gave the visitors a lead of 16 after they had quickly taken the final Leicestershire wicket this morning to dismiss their opponents for 320.