"No doubt they are being looked after in the proper place."

Where They Were Sent: Some First World War Asylums

Numerous questions were put to the government ministers by MPs concerned about the treatment of conscientious objectors (COs) suffering from mental illness, to be met with reassurances that such men were being treated reasonably and humanely (and that their condition was not a consequence of official policy towards them).

Questions in the House (selection)
HC Deb 28 November 1916 vol 88 cc133 Brutality cases Thomas Sidney Overbury
HC Deb 14 March 1917 vol 91 col.1109 Alfred Eungblut
HC Deb 13 November 1917 vol 99 cc193 Persecution, repeated CMs etc. John Taylor
HC Deb 05 February 1918 vol 101 cc2068 Wandsworth, Regulation 243A John Taylor
HC Deb 10 April 1918 vol 104 cc1461- Statistics; Bertie French
HC Deb 10 June 1918 vol 106 cc1852 Seventh Day Adventists; Le Havre; Harry Burgess
HC Deb 10 June 1918 vol 106 cc1885 William Stanton

The 'proper places' to which COs were sent generally bore no relation  to their place of origin. They were not clustered in a particular location, but distributed up and down the country. Another point to note is that as far as their onliue records go, none appears to have been sent to any asylum where a fellow CO was doing 'Work of National Importance' (WNI) as a member of the staff. 

Asylums with COS as patients
Cheshire County Asylum at Macclesfield
Denbigh Asylum
Derbyshire Asylum
Devon County Lunatic Asylum Exminster

Wakefield (engraving, 1818)

Richmond Asylum, Dublin
Dykebar War Hospital, Paisley
London County Asylum at Epsom 
Hanwell (4)
Hereford County Asylum
2nd Eastern Hospital, Hove
Ipswich
Lancaster Asylum (2)
Leicestershire County Lunatic Asylum
County Asylum at Rainhill, Liverpool
Morpeth County Asylum
West Riding Asylum, Wakefield (3) 

Some of the above had been taken over as 'military hospitals' so that COs could find themselves being kept/treated alongside psychiatric casualties. 

A few COs are reported to have been 'insane' in prison, their destination asylum, if any, not noted, e.g. Shepton Mallet CP;Winston Green CP (Civil Prison), Birmingham.

"In 1914 there were over one hundred thousand patients within over one hundred mental institutions around the United Kingdom, the majority of these institutions were built since the passing of the County Asylum / Lunacy Act in 1845. With the passing of the care in the community act in the 1980’s, many of these institutions have since closed; only a few of them remain open and in the use for Mental Health services." - https://www.countyasylums.co.uk/the-asylum-list/ 

'Hanwell' (London County) Asylum

"... 119 ‘County Asylums’ in both England and Wales. [Plus] additional asylums/hospitals which [may not] come under the ‘County Asylum’ list."

As to the treatment and improvement or deterioration in the condition of COs in asylums, the facts may still be difficult for researchers to ascertain generally, although individual records may be easier for descendants or relatives of patients to access: 

"Records of lunatic asylums are not held in any one place and often not all their records have survived. Many records of asylums, prisons and houses of correction are kept in local archives and especially those of the patients and inmates. However, most patient files have been destroyed."

Several of the asylums were considered to be putting into practice more humane treatments and regimen in the wake of nineteenth century concerns about the incarceration of the mentally ill; a few are seen as having been in the forefront of such reforms.

The humane imperative was not acknowledged everywhere or in all cases, however.
A comparison:
Endeavour. 2008 Dec;32(4):134-40. doi: 10.1016/j.endeavour.2008.09.001. Epub 2008 Nov 18.

Gags, funnels and tubes: forced feeding of the insane and of suffragettes.

Williams EA1.

Abstract

Just before the outbreak of World War I, British suffragettes were imprisoned in large numbers.
Many engaged in hunger strikes and suffered brutal treatment, most notoriously forced feeding.
Government authorities, backed by prominent physicians, justified forced feeding by citing
its successful use with insane patients in asylums.
In the nineteenth century forced feeding was, in fact, common in the asylum and much discussed
in leading medical publications.
Physicians generally ignored the feelings of patients, concentrating on technical problems
such as the design of feeding instruments.
Nor did critics amid the suffrage crisis sympathize with asylum patients.
They defended women protesters but portrayed the force-fed insane as insensate.
Forced feeding of the insane was nonetheless tainted by its association with
the brutalization of suffragettes and in later years rarely discussed
outside specialized psychiatric venues.

PMID: 19019439 DOI: 10.1016/j.endeavour.2008.09.001

[Reference: PubMed - indexed for MEDLINE]

(There are multiple points of similarity between government attempts to deal with 'absolutist' COs and the methods used against militant Suffragettes, especially when a hunger strike was involved.)