I don’t believe The Budget coverage has been on BBC1 since the The Budget itself was moved to lunchtime, which I believe changed when Tony Blair was PM. I know ITV had a couple of years off from covering it, but they’ve had it back for a good few years now.
The Budget hasn't regularly been on BBC1 since 1989! That was the year Parliament was televised for the first time and so there was Westminster coverage every week on BBC2, hence the Budget moved with it in 1990 and has been there ever since -
https://genome.ch.bbc.co.uk/schedules/bbctwo/england/1990-03-20#at-14.15
I think the only time it was on BBC1 since was in 1997 when it was during Wimbledon, so the Budget was on BBC1 and Wimbledon continued on BBC2, with no CBBC at all which seemed very strange at the time (though it was actually the second time that week it had happened, as there was no CBBC on the Monday of that week either with the Hong Kong Handover on BBC1). Of course it used to be that when the Budget was on BBC1, CBBC would move to BBC2, and although that stopped being the case in 1990, there would often be disruption to CBBC on Budget Day because, like this year, it was during the Cheltenham Festival and so CBBC would often start around 4.30.
As you say, ITV stopped showing it live for a bit in the nineties, seemingly they thought there was no point, and instead did an extended teatime news, which went under the banner of The Budget: What It Means To You. But they'd started showing it again by the time the teatime news moved to 6.30.
I would imagine most people get their Budget news from the One O'Clock News on BBC1, so between them the Beeb serves a wide audience. It's like what Radio 4 used to do in the eighties, they would have the full speech and the opposition response on FM, back when most of the audience listened on LW, and an extended edition of PM on LW dipping in and out of the speech and putting it in context.
Of course, the excitement of Budget Day when it was on in the afternoon was being able to tell your dad how much petrol had gone up by when he came home from work, the only thing any kid ever understood about it. It used to be a much bigger thing in those days, Radio 2 of course used to do a special programme with Jimmy Young, which used to be quite interesting because as JY was on in the afternoon, they had to shuffle around all the presenters to replace his usual show in the morning. And of course the Chancellor, plus the Shadow Chancellor and the Liberal/Alliance/Lib Dem finance spokesman would all have Budget Response Broadcasts on all channels for the next three nights. And there'd be a note in the Radio Times to say "Prices in advertisements may be affected by the Budget".