The letter he has sent staff is laughable to say the least. Drivers will only be able to TUPE over to other operators IF they complete remaining shifts.
Yes, that would be fairly standard. If you fail to complete your shifts you are considered to have quit without notice which would mean you are no longer eligible for employee benefits like transfer opportunities (TUPE), redundancy etc (which the more controlled manner of the closure would suggest may actually appear). You can also find, in these circumstances where you are moving to an operator who will be replacing work so don't need you until the after your last scheduled day, that if you walk out before the completing your last shifts you lose your new job offer as you have shown yourself to be unreliable, as operators will be taking over work it is in their interests for services to continue until they take over so a driver leaving early damages them as well as their current employer.
Probably, in reality, there won't be a formal TUPE process (and most new potential employers probably have better T&Cs in the round than Hulleys were offering so staying on existing T&Cs other than length of service may not be best) but you will go straight to the head of the queue in the application process and get assisted and streamlined re-employment process with all your details being transferred and set up so you can change job without a gap in pay/work that may accrue if you have to go through a standard recruitment procedures, and since you aren't legally TUPE'ing advertised signing on bonuses probably will still apply if you choose an operator who is offering such even if the transfer is arranged via your current employer.
With the drivers complaining about only getting 2 days notice, in circumstances like this that is actually quite good notice. I'm sure I remember when Tillingbourne went down in 2001 the drivers essentially got a call in the middle of the shift making them redundant and telling them to return to depot immediately and they were finished. A couple of days notice gives everyone a chance to sort things out, drivers have a chance to get a new position without a break (especially if the current employer is helping) & the council and other operators have a chance to arrange coverage.