I made an enquiry to HM Land Registry re the use of 0844 numbers and they referred me to the LR web information below: 

Call charges

Calls cost 3p a minute on a BT standard tariff, in addition to the current set up / connection charge. Calls from other tariffs, service providers and mobile phones may cost more. We do not receive any revenue from these calls.
Why doesn't Land Registry use an 0300 number?
Because of the way we would be charged to receive calls made under 0300, 0844 is currently the most cost effective option for this service, whilst not generating any revenue for Land Registry. However, we keep these arrangements under continual review.

My reply

Thank you for your prompt reply.  I have copied your web information below.  I do not understand the sentence "Because of the way we would be charged to receive calls made under 0300." can you explain to me what it means?
Yours
KK

The special features available to them on a non-geographic number have to be paid for by someone. Exactly the same features are available on all 03, 08 and 09 numbers.

On an 03 number, the number user has to pay for them. It's normally 1 or 2p/min, less if there is a high volume of calls.

On an 084, 087 or 09 number, the caller pays for these features through the Service Charge that is hidden within the inflated call price. The caller's phone provider also takes the opportunity to make some extra profit by further bumping up the call price. Organisations are tempted to use these numbers by the promise of no running costs and are told the call costs are low. The downsides are rarely mentioned.

If Land Registry continue using an 0844 number, they will shortly have to state the Service Charge (just over 3p/min) wherever their number is advertised.

Additionally, government policy is now based on "03 numbers":
http://www.gov.uk/government/publications/hmg-guidance-for-customer-service-lines

Reply from LR to my email.

Dear K,
Thank you for your further email dated 6 January 2014.
When our telephone support service was set up in 2009, other tariffs, for example 0300, were considered, however, the charge made to Land Registry to receive each call was higher than that for 0844 and was not consistent with Ofcom regulations that phone numbers should be ‘cost neutral'.
We do understand that things have since changed and as a result we have recently been reviewing our use of 0844  numbers. We are working to change our Customer Support number . However, due to the technical and infrastructure changes required to make this change I am not able to provide a precise date for change at the moment.
We will be announcing the change on our website as soon as we have a more definitive idea.
I hope this answers your query.
Yours sincerely
Alexandra Cook

Someone at LR has misunderstood what is meant by Cost Neutral.

03 numbers are cost neutral because the caller pays the price of a normal call and the user pays for the fancy call-handling features they want to have on their line.

0844 numbers load those running costs onto the caller, relieving the user of having to pay for them. 0844 numbers are not cost neutral.

This has always been the case, is still the case, and will continue to be the case. What is changing is how those costs are declared, thereby revealing the truth of the situation.

The technical changes are relatively simple. What's difficult is the logistics involved in communicating the new number in an effective manner and updating the many thousands of places where the old number is mentioned.

Many thanks for the link Ian.  I often ask questions to which I know the answer as a way of extracting extra information.

I'd be interested to know which "Ofcom regulations" Land Registry are referring to.

Whatever those regulations are, if indeed they exist, it is likely that LR has completely misunderstood the situation.

Ofcom encouraged public services to adopt 03 numbers, simply because they are cost neutral, as long ago as 2008.

It seems like LR misunderstood the memo, or are making excuses.