LHA rates are being revised and published each month. We have seen that some rates are going down from one month to the next, as well as some going up.

I can understand that, once the new rates are published, any new claims approved will be paid at those new rates.

However, do the new rates have any effect on existing claims? If they do, it appears that HB Depts. will have to review each existing claim every month & alter the amount paid (up or down) if the rate has changed. Or am I worrying unnecessarily and existing claims will go on at the rate when they were approved? If they do, how long is it before they are reviewed?

No, the LHA used on a claim lasts for a year unless there is a relevant change in circumstances. This is much the same way as the "old" Rent Officer arrangements worked.

Given that it's now confirmed that, with a weeks break and the usual process times, claimants can now ceae HB and move to LHA, is it going to be possible to cease LHA and move to a new claim on the current rates?

I have a suspicion that the housing situation may see substantial increases in rents over the next year or two that might make this a valid choice for some people.

Interesting question.
My initial answer is "I don't think so" but I need to think on it further.
The Regs says;

"it shall apply until the earlier of—

(i) the determination of a maximum rent (LHA) by virtue of regulation 13C(2)(d) (change of category of dwelling, death or change of dwelling for an LHA case);

(ii) the determination of a maximum rent (LHA) by virtue of regulation 13C(3) (anniversary of LHA date); or

(iii) the determination of a maximum rent by virtue of regulation 13 or an eligible rent under regulation 12B."

So a new claim does not appear to be one of the relevant changes, but I will need to look at things in more details to be certain.

Having looked at the guidance, I have changed my mind. The section on rapid reclaims says

"The rapid reclaim is treated as a new claim if there is a break in entitlement and this will trigger conversion to LHA or if already on LHA a new rate because it is a different claim"

Although we are not talking about a rapid reclaim here, the principal is the same so yes, I agree people can withdraw their claim to get a higher rate of LHA.

Jeff